🧠:<think>Alright, so I have this problem about analyzing school performance data in Andhra Pradesh. There are two sub-problems here. Let me try to tackle them one by one.Starting with Sub-problem 1: I need to calculate the weighted average score for the district. The weights are the number of students in each school. First, let me understand the given sequences. The number of students in each school is given by (a_n = 20 + 5n) for (n = 1) to (10). So, for each school from 1 to 10, the number of students increases by 5 each time. Similarly, the average math scores are given by (b_n = 60 + 2n). So, each school's average score increases by 2 points as n increases.To find the weighted average, I know the formula is the sum of (weight * value) divided by the sum of weights. In this case, the weights are the number of students (a_n), and the values are the average scores (b_n). So, the weighted average score (W) would be:[W = frac{sum_{n=1}^{10} a_n b_n}{sum_{n=1}^{10} a_n}]Alright, so I need to compute both the numerator and the denominator.Let me first compute the denominator, which is the total number of students in all schools. That is:[sum_{n=1}^{10} a_n = sum_{n=1}^{10} (20 + 5n)]This is an arithmetic series. Let me compute it term by term or find a formula. The sum of an arithmetic series is given by (S = frac{k}{2}(first term + last term)), where k is the number of terms.First, let's find the first term when n=1: (20 + 5*1 = 25).The last term when n=10: (20 + 5*10 = 20 + 50 = 70).So, the sum is (S = frac{10}{2}(25 + 70) = 5 * 95 = 475). So, total number of students is 475.Now, the numerator is the sum of (a_n b_n) for n=1 to 10.So, I need to compute (sum_{n=1}^{10} (20 + 5n)(60 + 2n)).Let me expand this product:[(20 + 5n)(60 + 2n) = 20*60 + 20*2n + 5n*60 + 5n*2n][= 1200 + 40n + 300n + 10n^2][= 1200 + 340n + 10n^2]So, the numerator becomes:[sum_{n=1}^{10} (1200 + 340n + 10n^2) = sum_{n=1}^{10} 1200 + sum_{n=1}^{10} 340n + sum_{n=1}^{10} 10n^2]Let me compute each part separately.First, (sum_{n=1}^{10} 1200): Since 1200 is constant, this is 1200*10 = 12,000.Second, (sum_{n=1}^{10} 340n): Factor out 340, so 340 * sum(n from 1 to 10). The sum of first 10 natural numbers is 55. So, 340*55 = let's compute that.340 * 50 = 17,000340 * 5 = 1,700So, total is 17,000 + 1,700 = 18,700.Third, (sum_{n=1}^{10} 10n^2): Factor out 10, so 10 * sum(n^2 from 1 to 10). The formula for the sum of squares is ( frac{n(n+1)(2n+1)}{6} ). For n=10:Sum = ( frac{10*11*21}{6} = frac{2310}{6} = 385 ).So, 10*385 = 3,850.Now, adding all three parts together:12,000 + 18,700 + 3,850 = 12,000 + 18,700 is 30,700; 30,700 + 3,850 is 34,550.So, the numerator is 34,550.Therefore, the weighted average score is:[W = frac{34,550}{475}]Let me compute that. 475 goes into 34,550 how many times?First, let me see: 475 * 70 = 33,250Subtract that from 34,550: 34,550 - 33,250 = 1,300Now, 475 * 2 = 950Subtract: 1,300 - 950 = 350475 goes into 350 zero times, but we can write it as 70 + 2 + (350/475). Simplify 350/475: divide numerator and denominator by 25: 14/19.So, W = 72 + 14/19 ≈ 72.7368.But maybe we can compute it more accurately.Wait, 475 * 72 = 475*(70 + 2) = 33,250 + 950 = 34,200.Subtract from 34,550: 34,550 - 34,200 = 350.So, 350/475 = 0.7368 approximately.So, W ≈ 72.7368.But perhaps we can write it as a fraction. 350/475 simplifies to 14/19, so W = 72 and 14/19, which is approximately 72.7368.So, the weighted average score is approximately 72.74.Wait, let me double-check my calculations because sometimes when dealing with large numbers, it's easy to make a mistake.First, the total number of students was 475, that's correct.The numerator: sum of (20 + 5n)(60 + 2n) from 1 to 10.We expanded it to 1200 + 340n + 10n², correct.Sum of 1200 ten times is 12,000.Sum of 340n is 340*55=18,700.Sum of 10n² is 10*(385)=3,850.Total numerator: 12,000 + 18,700 + 3,850 = 34,550. Correct.Divide 34,550 by 475:475*70=33,25034,550-33,250=1,300475*2=9501,300-950=350So, 70 + 2 =72, remainder 350.350/475=14/19≈0.7368So, 72.7368≈72.74. So, yes, that seems correct.So, the weighted average score is approximately 72.74.Moving on to Sub-problem 2: The educational officer wants to establish a performance index (P_n = frac{b_n}{a_n}) for each school. Then, he plans to allocate resources to the top 3 schools based on this index. I need to identify the top 3 schools and determine the total number of students in these schools.First, let's understand what (P_n) is. It's the average score divided by the number of students. So, higher (P_n) means either higher average score or fewer students, or both.Given that (a_n = 20 + 5n) and (b_n = 60 + 2n), so:[P_n = frac{60 + 2n}{20 + 5n}]We can simplify this expression:Factor numerator and denominator:Numerator: 2*(30 + n)Denominator: 5*(4 + n)So,[P_n = frac{2(30 + n)}{5(4 + n)} = frac{2}{5} * frac{30 + n}{4 + n}]Hmm, interesting. So, (P_n) is proportional to (frac{30 + n}{4 + n}). Let's see how this behaves as n increases.Let me compute (P_n) for each n from 1 to 10.Alternatively, maybe we can find a pattern or see whether (P_n) increases or decreases with n.Let me compute (P_n) for each school:For n=1:(P_1 = frac{60 + 2*1}{20 + 5*1} = frac{62}{25} = 2.48)n=2:(P_2 = frac{60 + 4}{20 + 10} = frac{64}{30} ≈ 2.1333)n=3:(P_3 = frac{60 + 6}{20 + 15} = frac{66}{35} ≈ 1.8857)n=4:(P_4 = frac{60 + 8}{20 + 20} = frac{68}{40} = 1.7)n=5:(P_5 = frac{60 + 10}{20 + 25} = frac{70}{45} ≈ 1.5556)n=6:(P_6 = frac{60 + 12}{20 + 30} = frac{72}{50} = 1.44)n=7:(P_7 = frac{60 + 14}{20 + 35} = frac{74}{55} ≈ 1.3455)n=8:(P_8 = frac{60 + 16}{20 + 40} = frac{76}{60} ≈ 1.2667)n=9:(P_9 = frac{60 + 18}{20 + 45} = frac{78}{65} ≈ 1.2)n=10:(P_{10} = frac{60 + 20}{20 + 50} = frac{80}{70} ≈ 1.1429)So, compiling these values:n | P_n---|---1 | 2.482 | ≈2.13333 | ≈1.88574 | 1.75 | ≈1.55566 | 1.447 | ≈1.34558 | ≈1.26679 | 1.210 | ≈1.1429So, clearly, as n increases, (P_n) decreases. That makes sense because both numerator and denominator increase, but the denominator increases faster relative to the numerator.Therefore, the highest (P_n) is at n=1, then n=2, then n=3, and so on.So, the top 3 schools are n=1, n=2, and n=3.Now, I need to determine the total number of students in these top 3 schools.Recall that (a_n = 20 + 5n).So, for n=1: (a_1 = 20 + 5*1 = 25)n=2: (a_2 = 20 + 10 = 30)n=3: (a_3 = 20 + 15 = 35)Total students = 25 + 30 + 35 = 90.So, the top 3 schools have a total of 90 students.Wait, let me double-check the calculations for (a_n):n=1: 20 +5=25n=2:20+10=30n=3:20+15=35Sum:25+30=55; 55+35=90. Correct.So, the top 3 schools are school 1, school 2, and school 3, with a total of 90 students.Just to make sure, let me cross-verify the performance index values:For n=1: 62/25=2.48n=2:64/30≈2.1333n=3:66/35≈1.8857Yes, these are the top three. So, correct.So, summarizing:Sub-problem 1: The weighted average score is approximately 72.74.Sub-problem 2: The top 3 schools are 1, 2, and 3, with a total of 90 students.Final AnswerSub-problem 1: The weighted average score is boxed{72.74}.Sub-problem 2: The total number of students in the top 3 schools is boxed{90}.</think>

🧠:<think>Alright, so I have this problem about a yoga instructor designing a prop shaped like a hyperbolic paraboloid. The equation given is ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ). The goal is to find the values of ( a ) and ( b ) such that the volume enclosed between this surface and the plane ( z = 0 ) is equal to a given constant ( V ). Then, using those values, calculate the maximum possible area of the shadow on the ( xy )-plane when illuminated from above.First, let me understand the problem step by step.1. Volume Calculation: The volume enclosed between the hyperbolic paraboloid and the plane ( z = 0 ) within the region ( -c leq x, y leq c ). So, this is a finite region, a square in the ( xy )-plane from ( -c ) to ( c ) in both x and y directions. The volume is given by integrating the function ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ) over this square region, but only where ( z ) is above the plane ( z = 0 ). Wait, but since it's a hyperbolic paraboloid, it has both positive and negative curvatures. So, depending on the values of ( x ) and ( y ), ( z ) can be positive or negative. However, the problem mentions the volume enclosed between the surface and the plane ( z = 0 ). So, I think that means we're considering the region where ( z geq 0 ). But wait, the hyperbolic paraboloid opens upwards along the x-axis and downwards along the y-axis. So, for positive ( x ), it curves up, and for positive ( y ), it curves down. So, the region where ( z geq 0 ) would be where ( frac{x^2}{a^2} geq frac{y^2}{b^2} ). That is, in the regions where ( |x| geq frac{a}{b} |y| ). But the problem states the region ( -c leq x, y leq c ). So, within this square, we have parts where ( z ) is positive and parts where ( z ) is negative. But the volume enclosed between the surface and ( z = 0 ) would be the integral of ( z ) over the region where ( z geq 0 ), right? Because where ( z ) is negative, the surface is below the plane, so the volume between them would be zero in those regions. Wait, but the problem says "the volume enclosed between the hyperbolic paraboloid ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ) and the plane ( z = 0 ) in the region ( -c leq x, y leq c )". So, perhaps it's considering the entire volume, both above and below the plane, but since volume can't be negative, we take the absolute value? Or maybe it's considering the net volume, but that might complicate things. Hmm, actually, the standard way to compute the volume between two surfaces is to integrate the difference between the upper surface and the lower surface. In this case, the hyperbolic paraboloid and the plane ( z = 0 ). So, if the paraboloid is above the plane in some regions and below in others, the volume would be the integral of the absolute difference. But that might complicate the integral. Alternatively, perhaps the problem is considering only the region where ( z geq 0 ), so the volume under the paraboloid above the plane. But then, the shape of the region where ( z geq 0 ) is not a square anymore, but a region bounded by ( |x| geq frac{a}{b} |y| ). So, within the square ( -c leq x, y leq c ), the area where ( z geq 0 ) is the region outside the lines ( x = pm frac{a}{b} y ). So, the volume would be the double integral over ( x ) and ( y ) from ( -c ) to ( c ), but only where ( z geq 0 ), of ( z , dx , dy ). Alternatively, maybe the problem is considering the entire volume, both above and below, but since volume is positive, it's the integral of ( |z| ) over the square. But that might not make much sense because the hyperbolic paraboloid is symmetric in a way, so the positive and negative volumes might cancel out if we just integrate ( z ). Wait, but the problem says "the volume enclosed between the hyperbolic paraboloid and the plane ( z = 0 )". So, I think it's the total volume between them, regardless of sign. So, that would be the integral of ( |z| ) over the region ( -c leq x, y leq c ). Hmm, but integrating ( |z| ) might complicate things because of the absolute value. Alternatively, maybe the problem is considering only the region where ( z geq 0 ), so we can compute the volume under the paraboloid above the plane within the square. Let me check the problem statement again: "the volume enclosed between the hyperbolic paraboloid ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ) and the plane ( z = 0 ) in the region ( -c leq x, y leq c )". So, it's the volume between the two surfaces in that region. So, if the paraboloid is above the plane in some parts and below in others, the enclosed volume would be the sum of the volumes where it's above and where it's below, but since volume is positive, it's the integral of ( |z| ). However, integrating ( |z| ) over the square might be complicated because of the absolute value. Alternatively, perhaps the problem assumes that the paraboloid is entirely above the plane within the region ( -c leq x, y leq c ). But given the equation ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ), unless ( frac{x^2}{a^2} geq frac{y^2}{b^2} ) for all ( x, y ) in ( -c leq x, y leq c ), which would require that ( frac{c^2}{a^2} geq frac{c^2}{b^2} ), so ( b geq a ). But the problem doesn't specify that, so I think we have to consider the absolute value. Alternatively, perhaps the problem is considering the volume where ( z geq 0 ), so the region where ( frac{x^2}{a^2} geq frac{y^2}{b^2} ), which is the area outside the lines ( y = pm frac{b}{a} x ). So, within the square ( -c leq x, y leq c ), the volume would be the integral over ( x ) from ( -c ) to ( c ), and for each ( x ), ( y ) ranges from ( - frac{a}{b} x ) to ( frac{a}{b} x ), but wait, no, actually, for each ( x ), the ( y ) where ( z geq 0 ) is ( |y| leq frac{b}{a} |x| ). So, the region is a diamond shape inside the square. Wait, that might complicate the integral because the limits of integration would depend on ( x ). Alternatively, maybe we can exploit symmetry. Let me try to visualize this. The hyperbolic paraboloid equation is ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ). So, along the x-axis, it's a parabola opening upwards, and along the y-axis, it's a parabola opening downwards. So, in the first quadrant, it's a saddle shape. The region ( -c leq x, y leq c ) is a square centered at the origin. The surface ( z ) is positive when ( frac{x^2}{a^2} > frac{y^2}{b^2} ), which is the region outside the lines ( y = pm frac{b}{a} x ). So, in each quadrant, the area where ( z geq 0 ) is a region bounded by those lines. Therefore, the volume would be four times the integral in the first quadrant where ( z geq 0 ). So, in the first quadrant, ( x ) goes from 0 to ( c ), and for each ( x ), ( y ) goes from 0 to ( frac{b}{a} x ). Wait, no, because ( z geq 0 ) when ( y leq frac{b}{a} x ), so in the first quadrant, ( y ) ranges from 0 to ( frac{b}{a} x ). So, the volume would be 4 times the integral from ( x = 0 ) to ( x = c ), and for each ( x ), ( y ) from 0 to ( frac{b}{a} x ), of ( z , dy , dx ). Let me write that down: ( V = 4 int_{0}^{c} int_{0}^{frac{b}{a} x} left( frac{x^2}{a^2} - frac{y^2}{b^2} right) dy , dx ) Alternatively, since the region is symmetric in all four quadrants, but the integrand is even in both x and y, so integrating over the entire region where ( z geq 0 ) would be four times the integral in the first quadrant. So, let's compute this integral step by step. First, compute the inner integral with respect to ( y ): ( int_{0}^{frac{b}{a} x} left( frac{x^2}{a^2} - frac{y^2}{b^2} right) dy ) Let's integrate term by term: ( int frac{x^2}{a^2} dy = frac{x^2}{a^2} y ) evaluated from 0 to ( frac{b}{a} x ), which is ( frac{x^2}{a^2} cdot frac{b}{a} x = frac{b x^3}{a^3} ) Next term: ( int frac{y^2}{b^2} dy = frac{1}{b^2} cdot frac{y^3}{3} ) evaluated from 0 to ( frac{b}{a} x ), which is ( frac{1}{b^2} cdot frac{(frac{b}{a} x)^3}{3} = frac{1}{b^2} cdot frac{b^3 x^3}{3 a^3} = frac{b x^3}{3 a^3} ) So, putting it together: ( int_{0}^{frac{b}{a} x} left( frac{x^2}{a^2} - frac{y^2}{b^2} right) dy = frac{b x^3}{a^3} - frac{b x^3}{3 a^3} = frac{2 b x^3}{3 a^3} ) Now, the outer integral becomes: ( V = 4 int_{0}^{c} frac{2 b x^3}{3 a^3} dx = 4 cdot frac{2 b}{3 a^3} int_{0}^{c} x^3 dx ) Compute the integral: ( int_{0}^{c} x^3 dx = frac{c^4}{4} ) So, ( V = 4 cdot frac{2 b}{3 a^3} cdot frac{c^4}{4} = frac{2 b c^4}{3 a^3} ) Therefore, the volume is ( V = frac{2 b c^4}{3 a^3} ). Wait, but the problem says the volume is equal to a given constant ( V ). So, we have: ( frac{2 b c^4}{3 a^3} = V ) So, we can solve for one variable in terms of the other. Let's say we solve for ( b ): ( b = frac{3 V a^3}{2 c^4} ) But we have two variables, ( a ) and ( b ), and only one equation. So, we need another condition to find both ( a ) and ( b ). The problem doesn't mention any other constraints, but perhaps the height ( h ) is given? Wait, the problem mentions "the prop has a finite height defined by ( z = h )". So, perhaps the maximum height of the prop is ( h ). So, the maximum value of ( z ) occurs at the edges of the region where ( z ) is maximum. Since the hyperbolic paraboloid opens upwards along x and downwards along y, the maximum ( z ) occurs at the maximum ( x ) and minimum ( y ). So, at ( x = c ), ( y = 0 ), ( z = frac{c^2}{a^2} ). Similarly, at ( x = -c ), ( y = 0 ), ( z = frac{c^2}{a^2} ). So, the maximum height ( h ) is ( frac{c^2}{a^2} ). Therefore, ( h = frac{c^2}{a^2} ) => ( a^2 = frac{c^2}{h} ) => ( a = frac{c}{sqrt{h}} ) So, now we can substitute ( a ) into the volume equation to find ( b ). From earlier, ( V = frac{2 b c^4}{3 a^3} ) Substitute ( a = frac{c}{sqrt{h}} ): ( V = frac{2 b c^4}{3 left( frac{c}{sqrt{h}} right)^3 } = frac{2 b c^4}{3 cdot frac{c^3}{h^{3/2}}} } = frac{2 b c^4 cdot h^{3/2}}{3 c^3} = frac{2 b c h^{3/2}}{3} ) So, ( V = frac{2 b c h^{3/2}}{3} ) Solving for ( b ): ( b = frac{3 V}{2 c h^{3/2}} ) Therefore, we have expressions for both ( a ) and ( b ) in terms of ( V ), ( c ), and ( h ). So, summarizing: ( a = frac{c}{sqrt{h}} ) ( b = frac{3 V}{2 c h^{3/2}} ) Let me check the units to see if they make sense. Volume ( V ) has units of length cubed, ( c ) is length, ( h ) is length. So, ( a ) is length, ( b ) is length. The expressions seem dimensionally consistent. Let me also check the logic. The maximum height is at ( x = c ), ( y = 0 ), so ( z = c^2 / a^2 = h ), so ( a = c / sqrt{h} ). That makes sense. Then, plugging ( a ) into the volume expression, we get ( b ) in terms of ( V ), ( c ), and ( h ). So, that seems correct. So, that answers part 1: ( a = frac{c}{sqrt{h}} ) and ( b = frac{3 V}{2 c h^{3/2}} ). Now, moving on to part 2: Calculate the maximum possible area of the prop's shadow on the ( xy )-plane for the region ( -c leq x, y leq c ) when the light source is directly above the prop. Hmm, the shadow on the ( xy )-plane when illuminated from above. Since the light is directly above, the shadow would be the projection of the prop onto the ( xy )-plane. But the prop is a 3D object, so its shadow would be the region on the ( xy )-plane where the prop blocks the light. However, since the prop is defined by ( z = frac{x^2}{a^2} - frac{y^2}{b^2} ), the shadow would be the projection of the surface onto the ( xy )-plane. But since the light is directly above, the shadow is just the region where the surface is present, but in this case, the surface is a hyperbolic paraboloid, which extends infinitely, but within the region ( -c leq x, y leq c ). Wait, but the shadow would be the projection of the entire prop, which is the region where ( z geq 0 ) within ( -c leq x, y leq c ). Because the prop is above the plane ( z = 0 ) in that region, so the shadow would be the area where the prop is present, which is the region ( |y| leq frac{b}{a} |x| ) within ( -c leq x, y leq c ). Wait, but the shadow is the projection, so actually, it's the area on the ( xy )-plane where the prop is blocking the light. Since the light is directly above, the shadow is the set of points ( (x, y) ) such that the line from the light source (at infinity above) to ( (x, y) ) intersects the prop. But since the prop is a surface, the shadow would be the projection of the surface onto the ( xy )-plane. However, for a hyperbolic paraboloid, the projection onto the ( xy )-plane would be the entire region where the surface exists, but since it's a saddle shape, the projection might actually be the entire square ( -c leq x, y leq c ), because for every ( (x, y) ), there is a corresponding ( z ). Wait, no, because the projection of a surface onto the ( xy )-plane is the set of all ( (x, y) ) such that there exists a ( z ) where ( (x, y, z) ) is on the surface. Since the surface is defined for all ( x ) and ( y ), the projection is the entire ( xy )-plane. But in our case, the prop is limited to ( -c leq x, y leq c ), so the shadow would be the entire square ( -c leq x, y leq c ). But that seems contradictory because the shadow area is supposed to be maximized, but if it's already the entire square, how can it be maximized? Maybe I'm misunderstanding. Alternatively, perhaps the shadow is the projection of the surface where ( z geq 0 ), which is the region ( |y| leq frac{b}{a} |x| ). So, the shadow would be the area of this region within ( -c leq x, y leq c ). Wait, but if the light is directly above, the shadow would be the projection of the entire prop, which is the region where ( z geq 0 ), because the prop is above the plane ( z = 0 ) in that region, so it blocks the light. So, the shadow is the area on the ( xy )-plane where ( z geq 0 ), which is the region ( |y| leq frac{b}{a} |x| ) within ( -c leq x, y leq c ). So, the area of the shadow is the area of this region. To find the maximum possible area, we need to maximize this area with respect to ( a ) and ( b ), but given that ( a ) and ( b ) are already determined from part 1, perhaps the area is fixed. Wait, but the problem says "calculate the maximum possible area of the prop's shadow on the ( xy )-plane for the region ( -c leq x, y leq c ) when the light source is directly above the prop." So, perhaps the shadow area depends on the shape of the prop, and we need to maximize it. But in part 1, we already determined ( a ) and ( b ) based on the volume ( V ) and height ( h ). So, perhaps the shadow area is fixed once ( a ) and ( b ) are fixed, so we just need to compute it. Alternatively, maybe the shadow area can be maximized by choosing ( a ) and ( b ) such that the area is maximized, but subject to the volume constraint. So, perhaps we need to maximize the shadow area given the volume ( V ). Wait, the problem says "once the values of ( a ) and ( b ) are determined, calculate the maximum possible area of the prop's shadow...". So, perhaps after determining ( a ) and ( b ) from part 1, we can compute the shadow area. So, let's proceed with that. The shadow area is the area on the ( xy )-plane where ( z geq 0 ), which is the region ( |y| leq frac{b}{a} |x| ) within ( -c leq x, y leq c ). So, the area is the area of the region bounded by ( |y| leq frac{b}{a} |x| ) within the square ( -c leq x, y leq c ). This region is a diamond (a square rotated by 45 degrees) inscribed within the square ( -c leq x, y leq c ). The diamond's vertices are at ( (c, 0) ), ( (0, c) ), ( (-c, 0) ), and ( (0, -c) ), but scaled by the slope ( frac{b}{a} ). Wait, no. The lines ( y = pm frac{b}{a} x ) intersect the square ( -c leq x, y leq c ) at points where ( x = pm c ) and ( y = pm frac{b}{a} c ), and where ( y = pm c ) and ( x = pm frac{a}{b} c ). So, depending on the value of ( frac{b}{a} ), the diamond may intersect the square at different points. If ( frac{b}{a} leq 1 ), then the lines ( y = pm frac{b}{a} x ) intersect the square at ( x = pm c ), ( y = pm frac{b}{a} c ), which are within the square. So, the diamond is entirely within the square. If ( frac{b}{a} > 1 ), then the lines ( y = pm frac{b}{a} x ) would intersect the square at ( y = pm c ), ( x = pm frac{a}{b} c ), which are within the square. So, in either case, the area of the shadow is the area of the diamond within the square. The area of the diamond (which is a square rotated by 45 degrees) is given by ( 2 times text{side length}^2 ). But in this case, the diamond is bounded by the lines ( y = pm frac{b}{a} x ) and the square ( -c leq x, y leq c ). Alternatively, the area can be computed as the integral over ( x ) from ( -c ) to ( c ) of the length of ( y ) where ( |y| leq frac{b}{a} |x| ). So, the area ( A ) is: ( A = 4 times int_{0}^{c} frac{b}{a} x , dx ) Because in the first quadrant, ( y ) ranges from 0 to ( frac{b}{a} x ), and we multiply by 4 for all quadrants. So, ( A = 4 times frac{b}{a} times int_{0}^{c} x , dx = 4 times frac{b}{a} times frac{c^2}{2} = 2 frac{b}{a} c^2 ) So, the area of the shadow is ( A = 2 frac{b}{a} c^2 ) Now, from part 1, we have expressions for ( a ) and ( b ): ( a = frac{c}{sqrt{h}} ) ( b = frac{3 V}{2 c h^{3/2}} ) So, let's compute ( frac{b}{a} ): ( frac{b}{a} = frac{frac{3 V}{2 c h^{3/2}}}{frac{c}{sqrt{h}}} = frac{3 V}{2 c h^{3/2}} times frac{sqrt{h}}{c} = frac{3 V}{2 c^2 h} ) Therefore, the area ( A ) is: ( A = 2 times frac{3 V}{2 c^2 h} times c^2 = frac{3 V}{h} ) So, the maximum possible area of the shadow is ( frac{3 V}{h} ). Wait, that seems too straightforward. Let me double-check. We have ( A = 2 frac{b}{a} c^2 ), and ( frac{b}{a} = frac{3 V}{2 c^2 h} ), so multiplying by 2 c^2 gives ( 2 times frac{3 V}{2 c^2 h} times c^2 = frac{3 V}{h} ). Yes, that's correct. So, the area of the shadow is ( frac{3 V}{h} ). Therefore, the maximum possible area is ( frac{3 V}{h} ). Wait, but the problem says "calculate the maximum possible area of the prop's shadow...". So, is this the maximum? Or is there a way to make the shadow area larger? Given that ( a ) and ( b ) are determined by the volume and height, the shadow area is fixed as ( frac{3 V}{h} ). So, perhaps that's the maximum possible area given the constraints. Alternatively, if we were to maximize the shadow area without considering the volume, we could make ( frac{b}{a} ) as large as possible, but since ( a ) and ( b ) are constrained by the volume and height, the shadow area is fixed. So, I think the answer is ( frac{3 V}{h} ). Let me recap: 1. We found ( a ) and ( b ) in terms of ( V ), ( c ), and ( h ). 2. Then, we computed the shadow area as ( 2 frac{b}{a} c^2 ), which simplified to ( frac{3 V}{h} ). So, the maximum possible area is ( frac{3 V}{h} ). Therefore, the answers are: 1. ( a = frac{c}{sqrt{h}} ) and ( b = frac{3 V}{2 c h^{3/2}} ) 2. The maximum shadow area is ( frac{3 V}{h} )</think>

🧠:<think>Okay, so I have this problem about analyzing the scoring patterns of two futsal teams, Team A and Team B. Let me try to break it down step by step. I'm a bit new to this, so I might need to take it slowly and make sure I understand each part.Starting with Team A. They play 10 matches, and the number of goals they score in each match follows a Poisson distribution with a mean of 2.5 goals per match. The first thing I need to do is calculate the probability that Team A scores exactly 4 goals in a single match. Then, I have to find the probability that they score 25 or more goals across all 10 matches.Alright, for the first part, the Poisson distribution. I remember that the Poisson probability mass function is given by:P(X = k) = (λ^k * e^(-λ)) / k!Where λ is the average rate (mean), which in this case is 2.5 goals per match. So, for exactly 4 goals, k is 4.Let me plug in the numbers:P(X = 4) = (2.5^4 * e^(-2.5)) / 4!First, I need to calculate 2.5^4. Let me compute that:2.5^2 = 6.25, so 2.5^4 = (2.5^2)^2 = 6.25^2 = 39.0625.Then, e^(-2.5). I know e is approximately 2.71828. So, e^(-2.5) is 1 / e^(2.5). Let me compute e^2.5.e^2 is about 7.389, and e^0.5 is about 1.6487. So, e^2.5 = e^2 * e^0.5 ≈ 7.389 * 1.6487 ≈ 12.1825. Therefore, e^(-2.5) ≈ 1 / 12.1825 ≈ 0.0821.Next, 4! is 4 factorial, which is 4*3*2*1 = 24.Putting it all together:P(X = 4) = (39.0625 * 0.0821) / 24First, multiply 39.0625 by 0.0821:39.0625 * 0.0821 ≈ Let's see, 39 * 0.08 is 3.12, and 39 * 0.0021 is approximately 0.0819. So total is roughly 3.12 + 0.0819 ≈ 3.2019.But more accurately, 39.0625 * 0.0821:Let me compute 39 * 0.0821 = 3.1999, and 0.0625 * 0.0821 ≈ 0.00513125. So total is approximately 3.1999 + 0.00513125 ≈ 3.20503125.So, approximately 3.205.Then, divide by 24: 3.205 / 24 ≈ 0.1335.So, approximately 0.1335, or 13.35%.Wait, let me double-check that calculation because sometimes I make arithmetic mistakes.Alternatively, maybe I can use a calculator approach:2.5^4 = 39.0625e^(-2.5) ≈ 0.082085Multiply them: 39.0625 * 0.082085 ≈ Let's compute 39 * 0.082085 = 3.199315, and 0.0625 * 0.082085 ≈ 0.0051303125. So total is approximately 3.199315 + 0.0051303125 ≈ 3.204445.Divide by 24: 3.204445 / 24 ≈ 0.1335185.So, approximately 0.1335, which is 13.35%.So, the probability that Team A scores exactly 4 goals in a single match is approximately 13.35%.Okay, that seems reasonable. Now, moving on to the second part: the probability that Team A scores a total of 25 or more goals across all 10 matches.Hmm, so this is about the sum of 10 independent Poisson random variables. I remember that the sum of independent Poisson variables is also Poisson, with the mean being the sum of the individual means.So, if each match has a mean of 2.5 goals, over 10 matches, the total mean would be 10 * 2.5 = 25 goals.So, the total number of goals, let's call it S, follows a Poisson distribution with λ = 25.We need to find P(S ≥ 25). That is, the probability that the total goals are 25 or more.But wait, calculating this directly might be tricky because the Poisson distribution is discrete, and calculating the sum from 25 to infinity can be cumbersome. Maybe we can use the normal approximation to the Poisson distribution since λ is reasonably large (25).I remember that for large λ, the Poisson distribution can be approximated by a normal distribution with mean μ = λ and variance σ² = λ.So, μ = 25, σ = sqrt(25) = 5.Therefore, S ~ N(25, 25).We need to find P(S ≥ 25). But since we're approximating a discrete distribution with a continuous one, we should apply a continuity correction.So, P(S ≥ 25) is approximately P(S ≥ 24.5) in the normal distribution.So, let's compute the z-score for 24.5.z = (24.5 - μ) / σ = (24.5 - 25) / 5 = (-0.5) / 5 = -0.1So, z = -0.1We need to find P(Z ≥ -0.1). Since the normal distribution is symmetric, this is equal to 1 - P(Z < -0.1).Looking up the standard normal distribution table, P(Z < -0.1) is approximately 0.4602.Therefore, P(Z ≥ -0.1) = 1 - 0.4602 = 0.5398.So, approximately 53.98%.But wait, let me think again. The exact probability for Poisson(25) that S ≥ 25. Since the mean is 25, the distribution is symmetric around 25? Wait, no, Poisson is not symmetric. It's skewed, but for large λ, it approximates a normal distribution which is symmetric.But actually, for Poisson, the distribution is skewed to the right, but with large λ, the skewness decreases.But in any case, using the normal approximation, we get approximately 53.98%.But let me check if I can compute the exact probability using Poisson.Calculating P(S ≥ 25) exactly would require summing from k=25 to infinity of (25^k * e^(-25)) / k!.But that's a lot of terms. Maybe we can use the complement: 1 - P(S ≤ 24).But even so, calculating P(S ≤ 24) is still a lot.Alternatively, maybe we can use the normal approximation with continuity correction as I did before.But let me see if I can get a better approximation or perhaps use another method.Alternatively, since the mean is 25, the median is around 25 as well for Poisson, so the probability of being above the mean is roughly 0.5. But with the continuity correction, it's slightly more than 0.5.Wait, but in the normal approximation, we found it to be approximately 0.5398, which is about 54%, which makes sense because the continuity correction shifts it slightly.Alternatively, maybe I can use the fact that for Poisson, the probability of being greater than or equal to the mean is roughly 0.5, but with some correction.But I think the normal approximation is acceptable here, especially since λ is 25, which is reasonably large.So, I think the approximate probability is about 53.98%, which we can round to approximately 54%.But let me see if I can get a better estimate.Alternatively, maybe using the Central Limit Theorem, since we're summing 10 independent Poisson variables, each with mean 2.5, so the total is Poisson(25). So, the sum can be approximated by a normal distribution with mean 25 and variance 25, as I did before.So, I think the normal approximation is the way to go here.Therefore, the probability that Team A scores 25 or more goals across all 10 matches is approximately 54%.Wait, but let me double-check the z-score calculation.We have S ~ N(25, 25). We want P(S ≥ 25). With continuity correction, it's P(S ≥ 24.5).So, z = (24.5 - 25)/5 = -0.1.Looking up z = -0.1 in the standard normal table, the cumulative probability is 0.4602, so the area to the right is 1 - 0.4602 = 0.5398, which is approximately 53.98%.Yes, that seems correct.Alternatively, if I didn't use continuity correction, then P(S ≥ 25) would be P(S > 25) in the continuous case, which would be P(S ≥ 25) = 1 - P(S < 25) = 1 - Φ((25 - 25)/5) = 1 - Φ(0) = 1 - 0.5 = 0.5.But since we're approximating a discrete distribution, the continuity correction is necessary, so 0.5398 is a better approximation.Therefore, I think 54% is a reasonable answer.Now, moving on to Team B.Team B's scoring in each match is normally distributed with a mean of 3 goals and a standard deviation of 1.5 goals. Alex wants to know the probability that Team B scores more than 5 goals in a single match. Additionally, calculate the 95% confidence interval for the total number of goals scored by Team B over 10 matches.First, the probability that Team B scores more than 5 goals in a single match.Since the distribution is normal, we can standardize it and use the Z-table.Let me denote X as the number of goals in a single match. X ~ N(μ = 3, σ = 1.5).We need to find P(X > 5).To find this, we can compute the z-score:z = (X - μ) / σ = (5 - 3) / 1.5 = 2 / 1.5 ≈ 1.3333.So, z ≈ 1.3333.We need to find P(Z > 1.3333). This is equal to 1 - P(Z ≤ 1.3333).Looking up z = 1.33 in the standard normal table, the cumulative probability is approximately 0.9082.But since 1.3333 is a bit more than 1.33, let's see if we can get a more precise value.Alternatively, using linear interpolation between z=1.33 and z=1.34.From standard normal tables:z=1.33: 0.9082z=1.34: 0.9099The difference between z=1.33 and z=1.34 is 0.01 in z, and the cumulative probabilities differ by 0.9099 - 0.9082 = 0.0017.Our z is 1.3333, which is 1.33 + 0.0033. So, the fraction is 0.0033 / 0.01 = 0.33.Therefore, the cumulative probability at z=1.3333 is approximately 0.9082 + 0.33*0.0017 ≈ 0.9082 + 0.000561 ≈ 0.908761.So, approximately 0.9088.Therefore, P(Z > 1.3333) = 1 - 0.9088 = 0.0912, or 9.12%.So, the probability that Team B scores more than 5 goals in a single match is approximately 9.12%.Alternatively, using a calculator or more precise z-table, the exact value might be slightly different, but 9.12% is a reasonable approximation.Now, the second part: calculate the 95% confidence interval for the total number of goals scored by Team B over 10 matches.So, we're dealing with the sum of 10 independent normal random variables. The sum of normals is also normal, with mean equal to the sum of the means and variance equal to the sum of the variances.Each match, the goals are N(3, 1.5^2). So, over 10 matches, the total goals, let's call it T, will be N(10*3, 10*(1.5)^2) = N(30, 22.5).So, T ~ N(30, 22.5). The standard deviation is sqrt(22.5) ≈ 4.7434.We need a 95% confidence interval for the total goals. Since we're dealing with a normal distribution, the confidence interval will be:μ ± z*(σ / sqrt(n))Wait, no, hold on. Wait, T is the total, which is already a sum. So, the standard error for the total is σ_total = sqrt(10)*σ_single = sqrt(10)*1.5 ≈ 4.7434, which is what I have above.But actually, for the confidence interval of the total, it's just μ_total ± z*(σ_total).Because T is a single normal variable with mean 30 and standard deviation sqrt(22.5). So, the 95% confidence interval is:30 ± z_{0.975} * sqrt(22.5)z_{0.975} is the critical value for a 95% confidence interval, which is 1.96.So, let's compute that.First, compute sqrt(22.5):sqrt(22.5) ≈ 4.7434.Then, 1.96 * 4.7434 ≈ Let's compute 2 * 4.7434 = 9.4868, so 1.96 is slightly less than 2, so 1.96 * 4.7434 ≈ 9.314.Wait, let me compute it more accurately:4.7434 * 1.96:First, 4 * 1.96 = 7.840.7434 * 1.96:Compute 0.7 * 1.96 = 1.3720.0434 * 1.96 ≈ 0.085So, total is 1.372 + 0.085 ≈ 1.457So, total is 7.84 + 1.457 ≈ 9.297.So, approximately 9.297.Therefore, the 95% confidence interval is:30 ± 9.297Which is approximately (30 - 9.297, 30 + 9.297) = (20.703, 39.297)So, the 95% confidence interval for the total number of goals scored by Team B over 10 matches is approximately (20.7, 39.3).But let me double-check the calculations.First, the total variance is 10*(1.5)^2 = 10*2.25 = 22.5, so standard deviation is sqrt(22.5) ≈ 4.7434.z_{0.975} is 1.96.So, 1.96 * 4.7434 ≈ Let me compute 4.7434 * 2 = 9.4868, so 4.7434 * 1.96 = 4.7434*(2 - 0.04) = 9.4868 - (4.7434*0.04) = 9.4868 - 0.1897 ≈ 9.2971.Yes, so 9.2971.Therefore, the confidence interval is 30 ± 9.2971, which is (20.7029, 39.2971). So, approximately (20.7, 39.3).Alternatively, if we want to be more precise, we can write it as (20.70, 39.30).So, summarizing:1. For Team A: - Probability of exactly 4 goals in a match: ~13.35% - Probability of 25 or more goals in 10 matches: ~54%2. For Team B: - Probability of more than 5 goals in a match: ~9.12% - 95% confidence interval for total goals in 10 matches: (20.7, 39.3)I think that covers all the parts of the problem. Let me just make sure I didn't miss anything.Wait, for Team A's second part, I approximated using the normal distribution. Is there another way to calculate it more accurately? Maybe using the Poisson cumulative distribution function.But since λ is 25, calculating the exact probability would require summing from 25 to infinity, which is not practical by hand. So, the normal approximation is the standard approach here.Alternatively, maybe using the De Moivre-Laplace theorem, which is the same as the normal approximation.So, I think the approach is correct.Similarly, for Team B, the confidence interval is straightforward once we know the total is normal with mean 30 and standard deviation ~4.7434.So, I think I've covered everything.Final Answer1. The probability that Team A scores exactly 4 goals in a single match is boxed{0.1335} and the probability that they score 25 or more goals in 10 matches is boxed{0.5398}.2. The probability that Team B scores more than 5 goals in a single match is boxed{0.0912}, and the 95% confidence interval for their total goals in 10 matches is boxed{(20.7, 39.3)}.</think>

🧠:<think>Alright, so I've got this problem about Mr. Smith, a lawyer who prefers paper records but has two income sources: legal consulting and real estate investments. I need to figure out his total annual income from consulting and then determine the minimum hours he needs to work to reach a total income of at least 150,000. Let me break this down step by step.First, part 1: Calculate his total annual income from legal consulting. He works x hours per week and earns 200 per hour. He works 50 weeks a year. Hmm, okay, so that seems straightforward. I think I just need to multiply his hourly rate by the number of hours he works per week and then by the number of weeks he works in a year.So, the formula for annual income from consulting would be:Annual Consulting Income = (Hourly Rate) × (Hours per Week) × (Weeks per Year)Plugging in the numbers:Annual Consulting Income = 200 × x × 50Let me compute that. 200 multiplied by 50 is 10,000. So, it's 10,000 × x. So, his annual consulting income is 10,000x dollars. That seems right. So, part 1 is done.Now, moving on to part 2. He wants his total annual income from both consulting and real estate to be at least 150,000. So, I need to figure out the minimum x such that the sum of his consulting income and real estate income is ≥ 150,000.First, I need to find his annual income from real estate. The function given is I(t) = 5000 + 300 sin(πt/6), where t is the number of months since January. So, this is a monthly income function. To get the annual income, I need to sum this over 12 months, right?Wait, but is it? Or is it that each month, his income is 5000 + 300 sin(πt/6). So, over 12 months, we can model each month's income and then sum them up.Alternatively, maybe we can compute the average monthly income and then multiply by 12. Let me think.The function I(t) = 5000 + 300 sin(πt/6). Let's analyze this function. The sine function oscillates between -1 and 1, so the 300 sin(πt/6) part will oscillate between -300 and +300. Therefore, the monthly income will vary between 5000 - 300 = 4700 and 5000 + 300 = 5300.But to find the total annual income, we need to sum I(t) from t = 1 to t = 12.Alternatively, since the sine function has a period, maybe we can compute the sum over one period.Wait, let's compute the period of the sine function. The general sine function is sin(Bt), where the period is 2π / B. In this case, B is π/6, so the period is 2π / (π/6) = 12. So, the period is 12 months. That means the function completes one full cycle over a year.Therefore, the sum over t = 1 to t = 12 of I(t) would be the sum of 5000 + 300 sin(πt/6) for each month.So, let's compute that.Sum of I(t) from t=1 to t=12 = Sum_{t=1}^{12} [5000 + 300 sin(πt/6)] = Sum_{t=1}^{12} 5000 + Sum_{t=1}^{12} 300 sin(πt/6)Compute each part separately.First, Sum_{t=1}^{12} 5000 = 5000 × 12 = 60,000.Second, Sum_{t=1}^{12} 300 sin(πt/6). Let's factor out the 300: 300 × Sum_{t=1}^{12} sin(πt/6).Now, we need to compute Sum_{t=1}^{12} sin(πt/6).Let me compute each term:t = 1: sin(π/6) = 0.5t = 2: sin(2π/6) = sin(π/3) ≈ 0.8660t = 3: sin(3π/6) = sin(π/2) = 1t = 4: sin(4π/6) = sin(2π/3) ≈ 0.8660t = 5: sin(5π/6) = 0.5t = 6: sin(6π/6) = sin(π) = 0t = 7: sin(7π/6) = -0.5t = 8: sin(8π/6) = sin(4π/3) ≈ -0.8660t = 9: sin(9π/6) = sin(3π/2) = -1t = 10: sin(10π/6) = sin(5π/3) ≈ -0.8660t = 11: sin(11π/6) = -0.5t = 12: sin(12π/6) = sin(2π) = 0Now, let's list all these:t=1: 0.5t=2: ≈0.8660t=3: 1t=4: ≈0.8660t=5: 0.5t=6: 0t=7: -0.5t=8: ≈-0.8660t=9: -1t=10: ≈-0.8660t=11: -0.5t=12: 0Now, let's add them up step by step.Start with 0.5 (t=1)Add 0.8660: 0.5 + 0.8660 ≈ 1.3660Add 1: 1.3660 + 1 = 2.3660Add 0.8660: 2.3660 + 0.8660 ≈ 3.2320Add 0.5: 3.2320 + 0.5 = 3.7320Add 0: 3.7320Add -0.5: 3.7320 - 0.5 = 3.2320Add -0.8660: 3.2320 - 0.8660 ≈ 2.3660Add -1: 2.3660 - 1 = 1.3660Add -0.8660: 1.3660 - 0.8660 = 0.5Add -0.5: 0.5 - 0.5 = 0Add 0: 0So, the total sum of sin(πt/6) from t=1 to t=12 is 0.Wait, that's interesting. So, the sum of the sine terms over a full period is zero. That makes sense because the sine function is symmetric over its period.Therefore, Sum_{t=1}^{12} sin(πt/6) = 0.Therefore, the total real estate income over the year is 60,000 + 300 × 0 = 60,000.So, Mr. Smith's annual real estate income is 60,000.Wait, that seems a bit low considering the function peaks at 5300. But when we sum over the year, the sine components cancel out, so the average is 5000 per month, leading to 60,000 annually. That makes sense.So, his total annual income is Consulting Income + Real Estate Income = 10,000x + 60,000.He wants this total to be at least 150,000.So, set up the inequality:10,000x + 60,000 ≥ 150,000Let me solve for x.Subtract 60,000 from both sides:10,000x ≥ 150,000 - 60,00010,000x ≥ 90,000Divide both sides by 10,000:x ≥ 90,000 / 10,000x ≥ 9So, x must be at least 9 hours per week.Wait, but let me double-check my calculations.First, annual consulting income: 200 * x * 50 = 10,000x. Correct.Annual real estate income: 60,000. Correct.Total income: 10,000x + 60,000 ≥ 150,000.Subtract 60,000: 10,000x ≥ 90,000.Divide by 10,000: x ≥ 9.Yes, that seems correct.But wait, let me think again about the real estate income. The function I(t) = 5000 + 300 sin(πt/6). So, each month, his income is 5000 plus a sine term. When we sum over 12 months, the sine terms cancel out, so the total is 12 * 5000 = 60,000. So, that part is correct.Therefore, the calculation is correct. So, he needs to work at least 9 hours per week.But let me just think about the sine function again. Since the sine function over a full period sums to zero, so the average is 5000 per month, so 60,000 per year. Therefore, the real estate income is fixed at 60,000, regardless of the sine fluctuations.Therefore, the consulting income needs to make up the difference between 60,000 and 150,000, which is 90,000. Since consulting income is 10,000x, x needs to be 9.Wait, but let me check: 9 hours per week times 50 weeks is 450 hours. 450 hours times 200 per hour is 450 * 200 = 90,000. So, yes, that's correct.So, the minimum number of hours per week is 9.Therefore, the answers are:1. Total annual consulting income is 10,000x, but since x is variable, but in part 1, it's just asking for the total, which is 10,000x. Wait, no, part 1 says "Calculate Mr. Smith's total annual income from his legal consulting work if he works 50 weeks in a year." So, actually, part 1 is just 200 * x * 50, which is 10,000x. But wait, is x given? Wait, no, in part 1, it's just asking for the formula, right? Or is it expecting a numerical value?Wait, let me read part 1 again: "Calculate Mr. Smith's total annual income from his legal consulting work if he works 50 weeks in a year."Wait, so he works x hours per week, 50 weeks a year. So, the total annual income is 200 * x * 50 = 10,000x. So, the answer is 10,000x dollars. But maybe they want it expressed as a function or as a numerical value? Wait, but x is a variable, so unless x is given, it's 10,000x.Wait, but in part 2, we find x. So, perhaps in part 1, it's just expressing the formula, which is 10,000x.But let me check the original problem again."1. Calculate Mr. Smith's total annual income from his legal consulting work if he works 50 weeks in a year."So, it's just asking for the total annual income from consulting, given he works 50 weeks. So, the formula is 200 * x * 50 = 10,000x. So, the answer is 10,000x dollars.But in part 2, we find that x needs to be at least 9, so his consulting income would be 90,000, and real estate is 60,000, totaling 150,000.So, to answer part 1, it's 10,000x.But wait, maybe I misread. Let me check again.Wait, the problem says "Calculate Mr. Smith's total annual income from his legal consulting work if he works 50 weeks in a year."So, it's not asking for a formula, but rather, given that he works 50 weeks, what is his total annual income. But since x is his hours per week, it's 200 * x * 50 = 10,000x. So, unless x is given, we can't compute a numerical value. Wait, but in the problem statement, x is defined as the number of hours he works per week. So, in part 1, it's just asking for the expression in terms of x, which is 10,000x.Alternatively, maybe I'm overcomplicating. Let me see.Wait, perhaps part 1 is just asking for the formula, and part 2 is asking for the value of x. So, in part 1, the answer is 10,000x, and in part 2, x is 9.But let me make sure. Let me re-express part 1: "Calculate Mr. Smith's total annual income from his legal consulting work if he works 50 weeks in a year."So, he works 50 weeks, x hours per week, earning 200 per hour. So, total income is 200 * x * 50 = 10,000x. So, yes, that's the formula.Therefore, part 1 answer is 10,000x.But wait, in the problem statement, it says "Mr. Smith practices law for x hours per week." So, x is a variable, so part 1 is just expressing the formula in terms of x.So, to sum up:1. Total annual consulting income: 10,000x dollars.2. To achieve total annual income of at least 150,000, he needs x ≥ 9 hours per week.Therefore, the answers are:1. 10,000x2. x = 9But let me just make sure about the real estate income. I calculated it as 60,000 per year because the sine terms cancel out. But let me think again: is the function I(t) = 5000 + 300 sin(πt/6). So, each month, the income is 5000 plus a sine term. Over 12 months, the sine terms sum to zero, so total real estate income is 12 * 5000 = 60,000. So, that's correct.Therefore, the calculations are accurate.So, final answers:1. 10,000x2. 9 hours per week.But wait, in part 1, is it possible that they expect a numerical value? But since x isn't given, it must be in terms of x. So, yes, 10,000x.Alternatively, maybe I'm supposed to compute it as 200 * x * 50, which is 10,000x, so that's correct.Therefore, I think I've got it right.</think>