🧠:<think>Okay, so I have this problem about Mr. Thompson's shop. Let me try to figure it out step by step. First, the current shop is a rectangle with dimensions 30 meters by 20 meters. That means the area is 30 multiplied by 20, which is 600 square meters. Got that. Now, the shop needs to relocate to a new space that's a right-angled triangle, but it has to maintain the same area of 600 square meters. One of the legs of this triangle is fixed at 15 meters. I need to find the length of the hypotenuse and the other leg.Alright, so for a right-angled triangle, the area is (base * height)/2. In this case, one leg is 15 meters, so let's call that the base. The other leg will be the height. Let me denote the other leg as 'b'. So, the area is (15 * b)/2 = 600. Let me write that equation down:(15 * b)/2 = 600To solve for 'b', I can multiply both sides by 2:15 * b = 1200Then divide both sides by 15:b = 1200 / 15Calculating that, 1200 divided by 15 is 80. So, the other leg is 80 meters. Hmm, that seems quite long, but I guess it's possible in a triangle. Now, to find the hypotenuse. In a right-angled triangle, the hypotenuse can be found using the Pythagorean theorem: hypotenuse squared equals the sum of the squares of the other two sides. So, hypotenuse 'c' is sqrt(15^2 + 80^2).Let me compute that:15 squared is 225, and 80 squared is 6400. Adding them together: 225 + 6400 = 6625.So, the hypotenuse is sqrt(6625). Let me see, sqrt(6625) can be simplified. Let's factor 6625.Divide 6625 by 25: 6625 / 25 = 265. So, 6625 = 25 * 265. 265 can be broken down further: 265 = 5 * 53. So, 6625 = 25 * 5 * 53 = 125 * 53. Wait, 125 is 5^3, so sqrt(6625) = sqrt(25 * 265) = 5 * sqrt(265). Hmm, sqrt(265) is approximately 16.28, so 5 times that is about 81.4 meters. But maybe I should leave it in exact form unless they ask for a decimal.So, the hypotenuse is 5*sqrt(265) meters, which is approximately 81.4 meters. Wait, let me double-check my calculations. 15 squared is 225, 80 squared is 6400, adding them gives 6625. Square root of 6625 is indeed sqrt(25*265) = 5*sqrt(265). Yeah, that seems right.So, for part 1, the other leg is 80 meters, and the hypotenuse is 5*sqrt(265) meters.Moving on to part 2. Mr. Thompson wants to build a storage room inside the triangular space. The storage room will occupy 25% of the shop's total area. Since the shop is 600 square meters, 25% of that is 150 square meters. So, the storage room is a right-angled triangle with area 150 square meters.The storage room has one leg parallel and adjacent to the 15-meter leg of the shop. The length of this leg is 6 meters. So, similar to the shop's triangle, the storage room is a right-angled triangle with one leg 6 meters, and the other leg we need to find. Let's denote the other leg as 'd'.The area of the storage room is (6 * d)/2 = 150. So, setting up the equation:(6 * d)/2 = 150Multiply both sides by 2:6 * d = 300Divide both sides by 6:d = 50 meters.Wait, 50 meters? That seems quite long again, but considering the original triangle had an 80-meter leg, maybe it's proportionally similar.Now, to find the hypotenuse of the storage room. Using the Pythagorean theorem again: hypotenuse 'e' is sqrt(6^2 + 50^2).Calculating that:6 squared is 36, 50 squared is 2500. Adding them together: 36 + 2500 = 2536.So, the hypotenuse is sqrt(2536). Let me see if that can be simplified. 2536 divided by 4 is 634. So, sqrt(2536) = sqrt(4*634) = 2*sqrt(634). 634 can be broken down: 634 divided by 2 is 317, which is a prime number. So, sqrt(634) is irrational. Therefore, the hypotenuse is 2*sqrt(634) meters, which is approximately sqrt(634) is about 25.18, so 2 times that is approximately 50.36 meters.Wait, let me verify the area. If the storage room has legs 6 and 50, area is (6*50)/2 = 150, which is correct. So, that seems okay.But wait, I just realized something. The storage room is inside the shop's triangle, and it's a right-angled triangle with one leg parallel and adjacent to the 15-meter leg. So, does that mean the triangles are similar?Because if one leg is 6 meters, which is 6/15 = 0.4 times the original leg, then the other leg should also be scaled by 0.4. So, 0.4 times 80 is 32 meters. But wait, I just calculated it as 50 meters. That seems conflicting.Hmm, maybe I made a mistake. Let me think.If the triangles are similar, then the ratio of the areas would be the square of the ratio of corresponding sides. The storage room is 25% of the area, so the ratio of areas is 0.25. Therefore, the ratio of sides should be sqrt(0.25) = 0.5. So, each side should be half the length of the original.Wait, but the leg given is 6 meters, which is not half of 15 meters (which would be 7.5 meters). So, maybe they aren't similar? Or maybe the storage room is not similar?Wait, the problem says the storage room is a right-angled triangle with one leg parallel and adjacent to the 15-meter leg. So, it's similar in orientation but not necessarily similar in shape.So, perhaps the triangles are not similar, which is why the other leg is 50 meters instead of 32 meters. So, my initial calculation was correct because it's just another right-angled triangle with one leg 6 meters and area 150, so the other leg is 50 meters.But then, why does the problem mention that the storage room is parallel and adjacent? Maybe it's just to indicate the orientation, not necessarily similarity.So, perhaps my first answer is correct. The storage room has legs 6 meters and 50 meters, hypotenuse sqrt(2536) meters.Wait, but let me think again. If the storage room is inside the shop's triangle, and one leg is 6 meters, which is less than 15 meters, then the other leg must be less than 80 meters? But 50 is less than 80, so that's okay.Alternatively, maybe the storage room is placed such that the 6-meter leg is along the 15-meter leg, and the other leg is along the 80-meter leg, but scaled down. But since the area is 25%, which is 1/4, so if the legs are scaled by 1/2, the area would be 1/4. So, if the legs are 7.5 and 40, the area would be (7.5*40)/2 = 150, which is correct.But in the problem, it says the storage room has one leg parallel and adjacent to the 15-meter leg, with length 6 meters. So, maybe the 6 meters is along the 15-meter leg, but the other leg is not scaled proportionally.So, perhaps the storage room is a smaller triangle with one leg 6 meters along the 15-meter side, but the other leg is not scaled by the same factor. So, in that case, the other leg is 50 meters, which is longer than the scaled-down 40 meters.Wait, that seems contradictory because if the storage room is inside the shop's triangle, how can one leg be 6 meters along the 15-meter side, and the other leg be 50 meters, which is longer than the original 80-meter leg? That doesn't make sense because 50 is less than 80, so maybe it's possible.Wait, 50 is less than 80, so it's okay. So, the storage room is a triangle with legs 6 and 50, which is inside the larger triangle with legs 15 and 80. So, the 6-meter leg is along the 15-meter side, and the 50-meter leg is along the 80-meter side. So, that seems okay.But then, why is the area 25%? Because 6 is 40% of 15, and 50 is 62.5% of 80. So, the area ratio is (6/15)*(50/80) = (0.4)*(0.625) = 0.25, which is 25%. So, that makes sense. So, the area is 25% because the product of the ratios is 0.25.So, that means the storage room is not similar to the shop's triangle, but it's scaled differently along each axis. So, my initial calculation was correct.Therefore, the other leg is 50 meters, and the hypotenuse is sqrt(6^2 + 50^2) = sqrt(36 + 2500) = sqrt(2536) meters. Simplifying sqrt(2536), as I did earlier, is 2*sqrt(634). So, that's the exact value, approximately 50.36 meters.Wait, but let me check if 6*50/2 is indeed 150. Yes, 6*50 is 300, divided by 2 is 150. Correct.So, to summarize:1. The new shop space is a right-angled triangle with legs 15 meters and 80 meters, hypotenuse 5*sqrt(265) meters.2. The storage room is a right-angled triangle with legs 6 meters and 50 meters, hypotenuse 2*sqrt(634) meters.I think that's it. I don't see any mistakes in my calculations now.</think>

🧠:<think>Alright, so I have this problem about calculating the Pearson correlation coefficient and a probability for some WWII data. Let me try to break it down step by step. I'm a bit new to this, so I might make some mistakes, but I'll try to figure it out.First, let me understand the data given. There are two sequences: the number of Allied bombers deployed each month, which is ( {a_n} ), and the success rate of missions each month, ( {b_n} ). Both sequences are defined for 12 months, from n=1 to n=12.The number of bombers each month is given by ( a_n = 50 + 10n ). So, for each month, starting from 1, the number increases by 10. That makes sense; it's a linear increase. Let me write out the first few terms to get a sense:- For n=1: 50 + 10(1) = 60- For n=2: 50 + 10(2) = 70- ...- For n=12: 50 + 10(12) = 170So, the number of bombers goes from 60 to 170 over the 12 months, increasing by 10 each month.The success rate is given by ( b_n = frac{1}{2} + frac{n}{24} ). Let me compute the first few terms as well:- For n=1: 0.5 + 1/24 ≈ 0.5 + 0.0417 ≈ 0.5417- For n=2: 0.5 + 2/24 ≈ 0.5 + 0.0833 ≈ 0.5833- ...- For n=12: 0.5 + 12/24 = 0.5 + 0.5 = 1.0So, the success rate starts around 0.54 and goes up to 1.0 over the 12 months, increasing by approximately 0.0417 each month.Now, the first question is to calculate the Pearson correlation coefficient between these two sequences. Pearson's r measures the linear correlation between two variables. The formula is:[r = frac{sum (x_i - bar{x})(y_i - bar{y})}{sqrt{sum (x_i - bar{x})^2 sum (y_i - bar{y})^2}}]Where ( bar{x} ) and ( bar{y} ) are the means of the two variables.So, I need to compute the means of ( a_n ) and ( b_n ), then compute the numerator and denominator as per the formula.Let me start by computing the means.First, for ( a_n ):Since ( a_n = 50 + 10n ), this is an arithmetic sequence with first term 60 and last term 170. The mean of an arithmetic sequence is just the average of the first and last terms, so:[bar{a} = frac{60 + 170}{2} = frac{230}{2} = 115]Alternatively, I can compute it by summing all ( a_n ) and dividing by 12. Let's verify:The sum of an arithmetic sequence is ( frac{n}{2}(first + last) ). So, sum of ( a_n ) is ( frac{12}{2}(60 + 170) = 6 * 230 = 1380 ). Therefore, mean is 1380 / 12 = 115. Yep, same result.Now, for ( b_n ):( b_n = 0.5 + frac{n}{24} ). Let's compute the mean.Alternatively, since ( b_n ) is also a linear function of n, it's an arithmetic sequence as well. Let's see:First term (n=1): 0.5 + 1/24 ≈ 0.5417Last term (n=12): 0.5 + 12/24 = 1.0So, the mean is the average of the first and last terms:[bar{b} = frac{0.5417 + 1.0}{2} ≈ frac{1.5417}{2} ≈ 0.7708]Alternatively, compute the sum of ( b_n ) and divide by 12.Sum of ( b_n ) is sum from n=1 to 12 of (0.5 + n/24) = 12*0.5 + (1/24) * sum from n=1 to 12 of n.Sum from n=1 to 12 is (12)(13)/2 = 78.So, sum of ( b_n ) is 6 + (78)/24 = 6 + 3.25 = 9.25Therefore, mean ( bar{b} = 9.25 / 12 ≈ 0.7708 ). Same as before.Alright, so now I have the means: ( bar{a} = 115 ), ( bar{b} ≈ 0.7708 ).Next, I need to compute the numerator of Pearson's r, which is the sum of the products of the deviations from the mean for each month.So, for each month n, compute ( (a_n - bar{a})(b_n - bar{b}) ), then sum all these up.Similarly, the denominator is the square root of the product of the sum of squared deviations for ( a_n ) and the sum of squared deviations for ( b_n ).So, let me structure this.First, let's compute the deviations for each n.But since both ( a_n ) and ( b_n ) are linear functions of n, maybe there's a smarter way than computing each term individually. But since it's only 12 terms, maybe it's manageable.Alternatively, since both sequences are linear, their covariance and variances can be computed using properties of linear sequences.Wait, let me think. Since ( a_n = 50 + 10n ) and ( b_n = 0.5 + (n)/24 ), both are linear in n. So, the Pearson correlation between them should be 1, because they are both perfectly linear functions of n. But wait, let me check.Wait, Pearson's correlation measures linear association. Since both variables are linear functions of n, they are perfectly linearly related, so their correlation should be 1. But let me verify that.Wait, but let me compute it step by step to confirm.Alternatively, maybe it's exactly 1, but let me compute it.But just to make sure, let's compute the covariance and variances.First, let's compute the covariance between ( a_n ) and ( b_n ).Covariance is given by:[text{Cov}(a, b) = frac{1}{n-1} sum (a_i - bar{a})(b_i - bar{b})]But Pearson's r is covariance divided by the product of standard deviations.But since we have the entire population (12 months), we can use the population covariance and standard deviations.So, Pearson's r is:[r = frac{text{Cov}(a, b)}{sigma_a sigma_b}]Where ( sigma_a ) and ( sigma_b ) are the population standard deviations.But since both ( a_n ) and ( b_n ) are linear functions of n, their covariance should be equal to the product of their slopes times the variance of n.Wait, let me think.Let me denote n as the variable, which goes from 1 to 12.So, ( a_n = 50 + 10n ), so slope is 10.( b_n = 0.5 + (1/24)n ), so slope is 1/24.Therefore, the covariance between ( a_n ) and ( b_n ) is equal to the covariance between 10n and (1/24)n, which is 10*(1/24)*variance(n).Variance of n is the variance of the numbers 1 to 12.Similarly, variance of ( a_n ) is variance of 10n, which is 10^2 * variance(n).Variance of ( b_n ) is (1/24)^2 * variance(n).Therefore, Pearson's r is:Cov(a,b) / (sigma_a sigma_b) = [10*(1/24)*var(n)] / [10*sqrt(var(n)) * (1/24)*sqrt(var(n))] = [10*(1/24)*var(n)] / [10*(1/24)*var(n)] = 1.So, Pearson's r is 1. That makes sense because both variables are perfectly linear functions of n, so they are perfectly correlated.But let me verify this by computing it manually.Alternatively, let me compute the numerator and denominator.First, compute the numerator: sum of (a_n - mean_a)(b_n - mean_b).Since a_n = 50 + 10n, mean_a = 115.So, a_n - mean_a = 50 + 10n - 115 = 10n - 65.Similarly, b_n = 0.5 + n/24, mean_b ≈ 0.7708.So, b_n - mean_b = 0.5 + n/24 - 0.7708 ≈ n/24 - 0.2708.Therefore, the product (a_n - mean_a)(b_n - mean_b) ≈ (10n - 65)(n/24 - 0.2708).Let me compute this for each n from 1 to 12.Wait, that might take a while, but let me see if I can find a pattern or compute it more efficiently.Alternatively, since a_n and b_n are linear functions of n, their deviations from the mean are also linear functions of n. Therefore, the product is a quadratic function of n, and the sum over n=1 to 12 can be computed using summation formulas.Let me denote:Let’s denote:( a_n - bar{a} = 10n - 65 )( b_n - bar{b} = frac{n}{24} - 0.2708 )So, their product is:( (10n - 65)left(frac{n}{24} - 0.2708right) )Let me expand this:= ( 10n * frac{n}{24} - 10n * 0.2708 - 65 * frac{n}{24} + 65 * 0.2708 )Simplify each term:= ( frac{10}{24}n^2 - 2.708n - frac{65}{24}n + 17.598 )Simplify further:= ( frac{5}{12}n^2 - (2.708 + 2.7083)n + 17.598 )Wait, 65/24 is approximately 2.7083.So, combining the linear terms:= ( frac{5}{12}n^2 - (2.708 + 2.7083)n + 17.598 )Wait, 2.708 + 2.7083 is approximately 5.4163.So,= ( frac{5}{12}n^2 - 5.4163n + 17.598 )Now, the sum over n=1 to 12 of this expression is:Sum = ( frac{5}{12} sum n^2 - 5.4163 sum n + 17.598 sum 1 )We can compute each summation separately.First, ( sum n^2 ) from 1 to 12 is ( frac{12 * 13 * 25}{6} ). Wait, the formula is ( frac{n(n+1)(2n+1)}{6} ).So, for n=12:= ( frac{12 * 13 * 25}{6} ) = ( frac{12}{6} * 13 * 25 ) = 2 * 13 * 25 = 650.Next, ( sum n ) from 1 to 12 is ( frac{12*13}{2} = 78 ).And ( sum 1 ) from 1 to 12 is 12.So, plugging back in:Sum = ( frac{5}{12} * 650 - 5.4163 * 78 + 17.598 * 12 )Compute each term:First term: ( frac{5}{12} * 650 ≈ 5 * 54.1667 ≈ 270.8333 )Second term: 5.4163 * 78 ≈ Let's compute 5 * 78 = 390, 0.4163 * 78 ≈ 32.4654, so total ≈ 390 + 32.4654 ≈ 422.4654Third term: 17.598 * 12 ≈ 211.176Now, putting it all together:Sum ≈ 270.8333 - 422.4654 + 211.176 ≈First, 270.8333 - 422.4654 ≈ -151.6321Then, -151.6321 + 211.176 ≈ 59.5439So, the numerator is approximately 59.5439.Wait, but earlier I thought the correlation should be 1, which would mean the numerator should be equal to the denominator. Let me check if I made a mistake in the calculations.Wait, maybe I messed up the signs or the arithmetic. Let me recompute the sum step by step.First term: ( frac{5}{12} * 650 )Compute 650 / 12 ≈ 54.1667Then, 5 * 54.1667 ≈ 270.8333Second term: 5.4163 * 78Compute 5 * 78 = 3900.4163 * 78: 0.4 * 78 = 31.2, 0.0163 * 78 ≈ 1.2714, so total ≈ 31.2 + 1.2714 ≈ 32.4714So, total second term ≈ 390 + 32.4714 ≈ 422.4714Third term: 17.598 * 1217 * 12 = 204, 0.598 * 12 ≈ 7.176, so total ≈ 204 + 7.176 ≈ 211.176Now, sum = 270.8333 - 422.4714 + 211.176Compute 270.8333 - 422.4714 ≈ -151.6381Then, -151.6381 + 211.176 ≈ 59.5379So, approximately 59.5379.Now, compute the denominator.Denominator is sqrt( sum(a deviations squared) * sum(b deviations squared) )First, compute sum(a deviations squared):( sum (a_n - bar{a})^2 = sum (10n - 65)^2 )Similarly, ( sum (b_n - bar{b})^2 = sum left( frac{n}{24} - 0.2708 right)^2 )Let me compute these.First, sum(a deviations squared):= ( sum_{n=1}^{12} (10n - 65)^2 )Let me expand this:= ( sum_{n=1}^{12} (100n^2 - 1300n + 4225) )So, sum = 100 * sum(n^2) - 1300 * sum(n) + 4225 * 12We already know sum(n^2) = 650, sum(n) = 78, so:= 100*650 - 1300*78 + 4225*12Compute each term:100*650 = 65,0001300*78: 1000*78=78,000; 300*78=23,400; total=78,000 + 23,400=101,4004225*12: 4000*12=48,000; 225*12=2,700; total=48,000 + 2,700=50,700So, sum = 65,000 - 101,400 + 50,700 = (65,000 + 50,700) - 101,400 = 115,700 - 101,400 = 14,300So, sum(a deviations squared) = 14,300Now, compute sum(b deviations squared):= ( sum_{n=1}^{12} left( frac{n}{24} - 0.2708 right)^2 )Let me denote ( c_n = frac{n}{24} - 0.2708 )So, ( c_n = frac{n}{24} - frac{6.4992}{24} ) ≈ ( frac{n - 6.4992}{24} )But maybe it's easier to compute each term individually.Alternatively, expand the square:= ( sum_{n=1}^{12} left( frac{n^2}{24^2} - 2 * frac{n}{24} * 0.2708 + 0.2708^2 right) )= ( frac{1}{576} sum n^2 - frac{2 * 0.2708}{24} sum n + 12 * 0.2708^2 )Compute each term:First term: ( frac{1}{576} * 650 ≈ 650 / 576 ≈ 1.1284 )Second term: ( frac{2 * 0.2708}{24} * 78 ≈ (0.5416 / 24) * 78 ≈ (0.02257) * 78 ≈ 1.756 )Third term: 12 * (0.2708)^2 ≈ 12 * 0.0733 ≈ 0.880So, sum ≈ 1.1284 - 1.756 + 0.880 ≈1.1284 - 1.756 ≈ -0.6276-0.6276 + 0.880 ≈ 0.2524Wait, that seems very low. Let me check my calculations.Wait, maybe I made a mistake in the second term.Wait, the second term is:( - frac{2 * 0.2708}{24} * 78 )Compute 2 * 0.2708 = 0.5416Divide by 24: 0.5416 / 24 ≈ 0.02257Multiply by 78: 0.02257 * 78 ≈ 1.756So, the second term is -1.756Third term: 12 * (0.2708)^20.2708^2 ≈ 0.073312 * 0.0733 ≈ 0.880So, total sum ≈ 1.1284 - 1.756 + 0.880 ≈1.1284 + 0.880 = 2.00842.0084 - 1.756 ≈ 0.2524So, sum(b deviations squared) ≈ 0.2524Wait, that seems really small. Let me check if I did the expansion correctly.Wait, when I expanded ( (c_n)^2 ), I had:( frac{n^2}{24^2} - 2 * frac{n}{24} * 0.2708 + 0.2708^2 )Yes, that's correct.But let me compute it differently. Maybe compute each term individually.Compute ( c_n = frac{n}{24} - 0.2708 ) for each n, square it, and sum.Let me make a table:n | c_n = n/24 - 0.2708 | c_n^2---|-------------------|-------1 | 0.0417 - 0.2708 = -0.2291 | (-0.2291)^2 ≈ 0.05252 | 0.0833 - 0.2708 = -0.1875 | (-0.1875)^2 ≈ 0.03523 | 0.125 - 0.2708 = -0.1458 | (-0.1458)^2 ≈ 0.02124 | 0.1667 - 0.2708 = -0.1041 | (-0.1041)^2 ≈ 0.01085 | 0.2083 - 0.2708 = -0.0625 | (-0.0625)^2 ≈ 0.00396 | 0.25 - 0.2708 = -0.0208 | (-0.0208)^2 ≈ 0.00047 | 0.2917 - 0.2708 = 0.0209 | (0.0209)^2 ≈ 0.00048 | 0.3333 - 0.2708 = 0.0625 | (0.0625)^2 ≈ 0.00399 | 0.375 - 0.2708 = 0.1042 | (0.1042)^2 ≈ 0.010810 | 0.4167 - 0.2708 = 0.1459 | (0.1459)^2 ≈ 0.021311 | 0.4583 - 0.2708 = 0.1875 | (0.1875)^2 ≈ 0.035212 | 0.5 - 0.2708 = 0.2292 | (0.2292)^2 ≈ 0.0525Now, let's sum these squared terms:0.0525 + 0.0352 + 0.0212 + 0.0108 + 0.0039 + 0.0004 + 0.0004 + 0.0039 + 0.0108 + 0.0213 + 0.0352 + 0.0525Let me add them step by step:Start with 0.0525+0.0352 = 0.0877+0.0212 = 0.1089+0.0108 = 0.1197+0.0039 = 0.1236+0.0004 = 0.1240+0.0004 = 0.1244+0.0039 = 0.1283+0.0108 = 0.1391+0.0213 = 0.1604+0.0352 = 0.1956+0.0525 = 0.2481So, the sum is approximately 0.2481, which is close to the earlier 0.2524. The slight difference is due to rounding errors in the individual terms. So, sum(b deviations squared) ≈ 0.2481Now, the denominator is sqrt(14,300 * 0.2481)Compute 14,300 * 0.2481 ≈ 14,300 * 0.25 = 3,575, but since it's 0.2481, it's slightly less.Compute 14,300 * 0.2481:First, 14,300 * 0.2 = 2,86014,300 * 0.04 = 57214,300 * 0.0081 ≈ 14,300 * 0.008 = 114.4, plus 14,300 * 0.0001 = 1.43, so total ≈ 114.4 + 1.43 ≈ 115.83So, total ≈ 2,860 + 572 + 115.83 ≈ 3,547.83So, 14,300 * 0.2481 ≈ 3,547.83Therefore, denominator ≈ sqrt(3,547.83) ≈ 59.56Wait, because 59.56^2 ≈ 3,547. So, yes.So, denominator ≈ 59.56Now, numerator was approximately 59.54So, Pearson's r ≈ 59.54 / 59.56 ≈ 0.9997Which is approximately 1, as expected.So, the Pearson correlation coefficient is approximately 1.But let me check if I made any mistake in the calculations.Wait, when I computed the sum of (a_n - mean_a)(b_n - mean_b), I got approximately 59.54, and the denominator was approximately 59.56, so the ratio is almost 1.Therefore, the Pearson correlation coefficient is 1.But wait, in reality, since both sequences are perfectly linear functions of n, their correlation should be exactly 1. The slight discrepancy is due to rounding errors in the calculations. So, the answer is 1.Now, moving on to the second question.Henry wants to know the probability that in any randomly chosen month from this data set, the number of bombers deployed will be greater than 100 and the success rate of missions will be at least 0.75.So, we need to find the number of months where a_n > 100 and b_n ≥ 0.75, divided by the total number of months (12).First, let's find the months where a_n > 100.Given a_n = 50 + 10n > 100So, 10n > 50 => n > 5Since n is integer, n ≥ 6.So, months 6 to 12: that's 7 months.Now, find the months where b_n ≥ 0.75.Given b_n = 0.5 + n/24 ≥ 0.75So, n/24 ≥ 0.25 => n ≥ 6Because 0.25 * 24 = 6.So, n ≥ 6.Therefore, the months where both a_n > 100 and b_n ≥ 0.75 are the months where n ≥ 6, which is 7 months.Therefore, the number of favorable months is 7.Total number of months is 12.So, the probability is 7/12.Simplify 7/12 ≈ 0.5833But as a fraction, it's 7/12.So, the probability is 7/12.Wait, let me verify.For a_n > 100: n ≥ 6, so months 6-12: 7 months.For b_n ≥ 0.75: n ≥ 6, same months.So, the overlap is 7 months.Therefore, probability is 7/12.Yes, that seems correct.So, to summarize:1. Pearson correlation coefficient is 1.2. Probability is 7/12.But let me just double-check the b_n condition.b_n = 0.5 + n/24 ≥ 0.75So, n/24 ≥ 0.25 => n ≥ 6.Yes, because 6/24 = 0.25, so 0.5 + 0.25 = 0.75.So, for n=6, b_n=0.75, and for n>6, it's higher.Therefore, months 6-12 satisfy both conditions.So, 7 months.Therefore, probability is 7/12.I think that's correct.</think>

🧠:<think>Alright, so I have this problem about organizing a vegan festival, and I need to figure out the maximum number of servings of two meals, Meal A and Meal B, that can be prepared given the constraints on tofu and vegetables. Then, in the second part, I have to consider pamphlets as well. Let me try to break this down step by step.Starting with Sub-problem 1. I need to model this as a linear programming problem. I remember that linear programming involves maximizing or minimizing a linear objective function subject to linear constraints. Here, the goal is to maximize the number of servings, which I assume is the total number of meals, so that would be the objective function.Let me define the variables first. Let’s say:Let x = number of servings of Meal ALet y = number of servings of Meal BOur goal is to maximize the total servings, so the objective function would be:Maximize Z = x + yNow, the constraints are based on the resources available: tofu and vegetables.Meal A requires 3 units of tofu and 2 units of vegetables per serving.Meal B requires 4 units of tofu and 1 unit of vegetables per serving.Total tofu available is 500 units, and vegetables are 300 units.So, the constraints can be written as:For tofu: 3x + 4y ≤ 500For vegetables: 2x + y ≤ 300Additionally, we can't have negative servings, so:x ≥ 0y ≥ 0So, summarizing, the linear programming problem is:Maximize Z = x + ySubject to:3x + 4y ≤ 5002x + y ≤ 300x, y ≥ 0Now, I need to solve this using the simplex method. Hmm, I remember the simplex method involves setting up a tableau and then performing pivot operations until an optimal solution is reached.First, I should convert the inequalities into equalities by introducing slack variables. Let’s add slack variables s and t for the tofu and vegetables constraints respectively.So, the constraints become:3x + 4y + s = 5002x + y + t = 300And the objective function remains:Z = x + yWe can write the initial simplex tableau:| Basis | x | y | s | t | RHS ||-------|---|---|---|---|-----|| s | 3 | 4 | 1 | 0 | 500 || t | 2 | 1 | 0 | 1 | 300 || Z | -1| -1| 0 | 0 | 0 |The basis consists of the slack variables s and t. The RHS column represents the right-hand side of the equations.In the simplex method, we select the entering variable with the most negative coefficient in the Z row. Here, both x and y have coefficients of -1, so I can choose either. Let me choose x as the entering variable.Next, we determine the leaving variable by calculating the minimum ratio of RHS to the corresponding coefficient in the entering variable's column. So, for s: 500 / 3 ≈ 166.67, and for t: 300 / 2 = 150. The smaller ratio is 150, so t will leave the basis.Now, we perform the pivot operation. The pivot element is the intersection of the entering variable x and leaving variable t, which is 2 in the t row.We need to make the pivot element 1 by dividing the entire t row by 2:t row: 2/2 = 1, 1/2 = 0.5, 0/2 = 0, 1/2 = 0.5, 300/2 = 150So, the new t row becomes:1 | 0.5 | 0 | 0.5 | 150Now, we need to eliminate x from the other rows. Starting with the s row:Current s row: 3 | 4 | 1 | 0 | 500We need to subtract 3 times the new t row from the s row.Calculating:3 - 3*1 = 04 - 3*0.5 = 4 - 1.5 = 2.51 - 3*0 = 10 - 3*0.5 = -1.5500 - 3*150 = 500 - 450 = 50So, the new s row is:0 | 2.5 | 1 | -1.5 | 50Now, the Z row:Current Z row: -1 | -1 | 0 | 0 | 0We need to add 1 times the new t row to the Z row to eliminate x.Calculating:-1 + 1*1 = 0-1 + 1*0.5 = -0.50 + 1*0 = 00 + 1*0.5 = 0.50 + 1*150 = 150So, the new Z row is:0 | -0.5 | 0 | 0.5 | 150Now, the tableau looks like this:| Basis | x | y | s | t | RHS ||-------|---|----|----|----|-----|| s | 0 |2.5 |1 |-1.5|50 || x |1 |0.5 |0 |0.5 |150 || Z |0 |-0.5|0 |0.5 |150 |Now, looking at the Z row, the coefficient for y is -0.5, which is still negative. So, y is the entering variable.We need to determine the leaving variable. Calculate the minimum ratio of RHS to the y coefficient in each row.For s: 50 / 2.5 = 20For x: 150 / 0.5 = 300The smaller ratio is 20, so s will leave the basis.Pivot on the y column and s row. The pivot element is 2.5.First, make the pivot element 1 by dividing the s row by 2.5:s row: 0 / 2.5 = 0, 2.5 / 2.5 = 1, 1 / 2.5 = 0.4, -1.5 / 2.5 = -0.6, 50 / 2.5 = 20So, new s row:0 |1 |0.4 |-0.6 |20Now, eliminate y from the other rows.Starting with the x row:Current x row:1 |0.5 |0 |0.5 |150We need to subtract 0.5 times the new s row from the x row.Calculating:1 - 0.5*0 = 10.5 - 0.5*1 = 00 - 0.5*0.4 = -0.20.5 - 0.5*(-0.6) = 0.5 + 0.3 = 0.8150 - 0.5*20 = 150 - 10 = 140So, the new x row is:1 |0 |-0.2 |0.8 |140Now, the Z row:Current Z row:0 |-0.5 |0 |0.5 |150We need to add 0.5 times the new s row to the Z row.Calculating:0 + 0.5*0 = 0-0.5 + 0.5*1 = 00 + 0.5*0.4 = 0.20.5 + 0.5*(-0.6) = 0.5 - 0.3 = 0.2150 + 0.5*20 = 150 + 10 = 160So, the new Z row is:0 |0 |0.2 |0.2 |160Now, the tableau is:| Basis | x | y | s | t | RHS ||-------|---|---|----|----|-----|| y |0 |1 |0.4 |-0.6|20 || x |1 |0 |-0.2 |0.8 |140 || Z |0 |0 |0.2 |0.2 |160 |Now, looking at the Z row, all coefficients are non-negative (0.2 and 0.2), so we have reached optimality.Therefore, the optimal solution is:x = 140 servings of Meal Ay = 20 servings of Meal BTotal servings Z = 160.Wait, let me verify that with the constraints.Tofu: 3*140 + 4*20 = 420 + 80 = 500, which matches the available tofu.Vegetables: 2*140 + 1*20 = 280 + 20 = 300, which matches the available vegetables.Okay, that seems correct.Now, moving on to Sub-problem 2. The organizer wants to distribute pamphlets. Each Meal A includes 3 pamphlets, and each Meal B includes 2 pamphlets. Total pamphlets available are 600.We already have the optimal number of servings from Sub-problem 1: x = 140, y = 20.So, let's calculate the total pamphlets required:3*140 + 2*20 = 420 + 40 = 460 pamphlets.Since 460 ≤ 600, the pamphlet constraint is satisfied. So, we don't need to adjust the number of servings. The optimal solution remains x = 140, y = 20.But wait, just to be thorough, let me consider if the pamphlet constraint was not satisfied. Suppose, for example, that the pamphlets required were more than 600. Then, we would have to adjust the servings.But in this case, since 460 is less than 600, the constraint is satisfied. So, the optimal solution doesn't change.Alternatively, if we were to include the pamphlet constraint in the original problem, we would have another constraint:3x + 2y ≤ 600But since in the initial problem, the pamphlet constraint wasn't part of the optimization, and after checking, it's satisfied, so no adjustment is needed.Therefore, the optimal solution remains 140 servings of Meal A and 20 servings of Meal B.Final AnswerSub-problem 1: The maximum number of servings is boxed{140} for Meal A and boxed{20} for Meal B.Sub-problem 2: The pamphlet constraint is satisfied with the same number of servings, so no adjustment is needed. The optimal solution remains boxed{140} servings of Meal A and boxed{20} servings of Meal B.</think>

🧠:<think>Okay, so I have this problem where I need to optimize a Swift application. The original algorithm has a time complexity of ( T(n) = 5n^2 + 3n ), and after refactoring, it becomes ( T'(n) = an log n + bn ). The key point is that for a large dataset size ( n = 10^6 ), the refactored algorithm runs twice as fast as the original. I need to find the values of ( a ) and ( b ).First, I should understand what it means for the refactored algorithm to run twice as fast. If the original time is ( T(n) ), then the refactored time ( T'(n) ) should be half of that, right? So, ( T'(n) = frac{1}{2} T(n) ).Let me write that down:( an log n + bn = frac{1}{2} (5n^2 + 3n) )Now, I can plug in ( n = 10^6 ) into this equation. But before that, maybe I can simplify the equation by dividing both sides by ( n ), since ( n ) is non-zero. That would give:( a log n + b = frac{1}{2} (5n + 3) )Wait, is that correct? Let me check:Original equation:( an log n + bn = frac{1}{2} (5n^2 + 3n) )Divide both sides by ( n ):( a log n + b = frac{1}{2} (5n + 3) )Yes, that seems right.Now, plugging in ( n = 10^6 ):( a log(10^6) + b = frac{1}{2} (5 times 10^6 + 3) )I need to compute ( log(10^6) ). Assuming the logarithm is base 2, since in computer science, we usually use base 2 for time complexities. Let me verify:( log_2(10^6) ) is approximately ( log_2(10^6) ). Since ( 2^{20} ) is about a million, so ( log_2(10^6) approx 20 ).Wait, actually, ( 2^{20} = 1,048,576 ), which is roughly ( 10^6 ). So, ( log_2(10^6) approx 20 ).So, plugging that in:( a times 20 + b = frac{1}{2} (5,000,000 + 3) )Simplify the right side:( 5,000,000 + 3 = 5,000,003 ), so half of that is approximately ( 2,500,001.5 ).So, the equation becomes:( 20a + b = 2,500,001.5 )Hmm, that's one equation with two unknowns. I need another equation to solve for ( a ) and ( b ). But wait, the problem doesn't provide another condition. Maybe I need to consider the behavior as ( n ) becomes large. Since ( n ) is ( 10^6 ), which is quite large, perhaps the dominant terms are the ones with the highest growth rates.In the original time complexity, the dominant term is ( 5n^2 ), and in the refactored, it's ( an log n ). So, for large ( n ), the ( n^2 ) term will dominate over the ( n log n ) term. But since the refactored algorithm is supposed to run twice as fast, maybe the leading coefficients should satisfy a certain relationship.Wait, but in the equation we have, ( 20a + b = 2,500,001.5 ). If ( n ) is very large, the term ( an log n ) is much smaller than ( bn ) if ( a ) is small. But since ( n ) is ( 10^6 ), and ( log n ) is about 20, ( an log n ) is ( 20a times 10^6 ), which is 20 million ( a ). Similarly, ( bn ) is ( b times 10^6 ). So, both terms are on the order of millions.But in the original equation, the right side is ( 2,500,001.5 ), which is about 2.5 million. So, the left side is ( 20a + b ) equals approximately 2.5 million. So, ( 20a + b approx 2.5 times 10^6 ).But we have only one equation with two variables. Maybe we need to assume that the lower-order terms are negligible? Or perhaps the problem expects us to find ( a ) and ( b ) such that the equation holds for ( n = 10^6 ), without considering other values. But with only one equation, we can't find two variables uniquely. Maybe the problem expects us to express ( b ) in terms of ( a ), or vice versa.Wait, let me reread the problem. It says, "determine the values of ( a ) and ( b )." So, perhaps there's another condition or maybe we can assume that the lower-order terms are negligible? Or maybe the problem expects us to consider that for large ( n ), the ( n log n ) term dominates over the ( n ) term, so ( b ) can be considered negligible? But that might not be the case here because ( n log n ) is 20 million ( a ), and ( bn ) is 1 million ( b ). So, if ( a ) is, say, 125, then ( 20a = 2,500,000 ), and ( b ) would be about 1.5. So, ( b ) is much smaller than ( a ), but not negligible.Alternatively, maybe the problem expects us to consider that the original algorithm's time is dominated by ( 5n^2 ), so the refactored algorithm's time is half of that, so ( an log n ) should be roughly half of ( 5n^2 ). So, ( an log n = frac{5}{2} n^2 ). Then, ( a log n = frac{5}{2} n ). But that would lead to ( a = frac{5}{2} times frac{n}{log n} ), which would make ( a ) dependent on ( n ), but ( a ) is supposed to be a constant. So, that approach might not be correct.Wait, perhaps I need to think differently. The original time is ( 5n^2 + 3n ), and the refactored is ( an log n + bn ). The refactored runs twice as fast, meaning that ( T'(n) = frac{1}{2} T(n) ). So, for ( n = 10^6 ):( an log n + bn = frac{1}{2} (5n^2 + 3n) )We can write this as:( an log n + bn = frac{5}{2}n^2 + frac{3}{2}n )Then, rearranged:( an log n = frac{5}{2}n^2 + frac{3}{2}n - bn )Factor out ( n ):( an log n = frac{5}{2}n^2 + left( frac{3}{2} - b right) n )Divide both sides by ( n ):( a log n = frac{5}{2}n + left( frac{3}{2} - b right) )Now, plug in ( n = 10^6 ):( a times 20 = frac{5}{2} times 10^6 + left( frac{3}{2} - b right) )Simplify the right side:( frac{5}{2} times 10^6 = 2,500,000 )So,( 20a = 2,500,000 + frac{3}{2} - b )Which is:( 20a + b = 2,500,000 + frac{3}{2} )Which is approximately:( 20a + b = 2,500,001.5 )So, that's the same equation I had before. Since I have only one equation with two variables, I can't find unique values for ( a ) and ( b ). Maybe the problem expects me to express ( b ) in terms of ( a ), or perhaps there's an assumption that ( b ) is negligible compared to ( a log n ), but I don't think that's the case here because ( b ) is multiplied by ( n ), which is large.Alternatively, maybe the problem expects me to consider that for large ( n ), the ( an log n ) term is the dominant term in ( T'(n) ), so perhaps ( b ) can be considered as a lower-order term and thus set to zero? But that might not be accurate because ( bn ) is still a significant term when ( n ) is large.Wait, let me think about the original equation again:( an log n + bn = frac{1}{2} (5n^2 + 3n) )If I consider that ( n log n ) is much smaller than ( n^2 ), but in this case, the refactored algorithm is supposed to be faster, so ( an log n ) must be significantly smaller than ( 5n^2 ). So, perhaps the dominant term in ( T'(n) ) is ( an log n ), and the dominant term in ( T(n) ) is ( 5n^2 ). Therefore, to make ( T'(n) = frac{1}{2} T(n) ), the dominant terms must satisfy:( an log n = frac{5}{2} n^2 )So,( a = frac{5}{2} times frac{n}{log n} )But ( a ) is a constant, not dependent on ( n ). So, this approach might not work because ( a ) would vary with ( n ), which contradicts the definition of ( a ) being a constant.Hmm, maybe I need to consider that the entire expressions are equal, not just the dominant terms. So, let's go back to the equation:( 20a + b = 2,500,001.5 )Since I have only one equation, perhaps the problem expects me to express ( b ) in terms of ( a ), or vice versa. But the question asks for the values of ( a ) and ( b ), implying that they are specific constants. Maybe I need to make an assumption, like setting ( b ) to zero? But that might not be correct because ( bn ) is part of the time complexity.Alternatively, perhaps the problem expects me to ignore the ( bn ) term because it's a lower-order term compared to ( an log n ). But when ( n ) is ( 10^6 ), ( an log n ) is about 20 million ( a ), and ( bn ) is 1 million ( b ). So, if ( a ) is around 125, then ( 20a = 2,500,000 ), and ( b ) would be about 1.5. So, ( b ) is much smaller than ( a ), but not negligible.Wait, maybe the problem expects me to consider that the ( bn ) term is negligible compared to ( an log n ), so I can approximate ( 20a approx 2,500,000 ), leading to ( a approx 125,000 ). But that seems too large because ( a ) is a constant, and ( an log n ) would then be 125,000 * 10^6 * 20, which is 2.5 * 10^12, which is way larger than the original time complexity. That doesn't make sense because the refactored algorithm is supposed to be faster.Wait, no, the original time complexity is ( 5n^2 + 3n ), which for ( n = 10^6 ) is ( 5*(10^6)^2 + 3*10^6 = 5*10^{12} + 3*10^6 approx 5*10^{12} ). The refactored time is ( an log n + bn approx 20a*10^6 + b*10^6 ). Since the refactored time is half of the original, it should be ( 2.5*10^{12} ).So, ( 20a*10^6 + b*10^6 = 2.5*10^{12} )Divide both sides by ( 10^6 ):( 20a + b = 2.5*10^6 )Wait, that's different from what I had earlier. Earlier, I had ( 20a + b = 2,500,001.5 ), which is approximately ( 2.5*10^6 ). So, maybe I can write:( 20a + b = 2.5*10^6 )But again, with two variables, I can't solve for both ( a ) and ( b ) uniquely. Unless there's another condition or unless I'm supposed to assume that ( b ) is negligible, but that might not be accurate.Wait, perhaps the problem expects me to consider that the ( bn ) term is much smaller than ( an log n ), so I can approximate ( 20a approx 2.5*10^6 ), leading to ( a approx 125,000 ). But as I thought earlier, that would make ( an log n ) way too large. Wait, no, because ( a ) is multiplied by ( n log n ), which is 20 million. So, ( a = 125,000 ) would make ( an log n = 125,000 * 20,000,000 = 2.5*10^{12} ), which is exactly half of the original time complexity. So, in that case, ( b ) would be negligible because ( 20a = 2.5*10^6 ), so ( b ) would be ( 2.5*10^6 - 20a = 0 ). So, ( b = 0 ).Wait, that makes sense. If ( a = 125,000 ), then ( 20a = 2,500,000 ), and ( b = 0 ). So, the refactored time complexity is ( 125,000 n log n ). But wait, that would mean the time complexity is dominated by ( an log n ), and the ( bn ) term is zero. So, the equation becomes:( 125,000 n log n = frac{1}{2} (5n^2 + 3n) )But for ( n = 10^6 ), ( 125,000 * 10^6 * 20 = 2.5*10^{12} ), which is exactly half of the original time complexity ( 5*10^{12} ). So, that works.But wait, ( a = 125,000 ) seems very large. Is that reasonable? In terms of constants, they can be large, but maybe I made a mistake in the units. Let me check:Original time complexity: ( T(n) = 5n^2 + 3n ). For ( n = 10^6 ), ( T(n) = 5*(10^6)^2 + 3*10^6 = 5*10^{12} + 3*10^6 approx 5*10^{12} ).Refactored time complexity: ( T'(n) = an log n + bn ). For ( n = 10^6 ), ( T'(n) = a*10^6*20 + b*10^6 = 20a*10^6 + b*10^6 ). We want this to be half of the original, so ( 20a*10^6 + b*10^6 = 2.5*10^{12} ).Divide both sides by ( 10^6 ):( 20a + b = 2.5*10^6 )So, if I set ( b = 0 ), then ( 20a = 2.5*10^6 ), so ( a = 125,000 ). That seems correct mathematically, but in practice, having such a large constant might not be realistic, but since it's a theoretical problem, maybe it's acceptable.Alternatively, if I don't set ( b = 0 ), I can express ( b ) in terms of ( a ):( b = 2.5*10^6 - 20a )But without another condition, I can't determine unique values for ( a ) and ( b ). So, perhaps the problem expects me to assume that ( b = 0 ), making ( a = 125,000 ).Wait, but the original time complexity has a ( 3n ) term, so the refactored algorithm's ( bn ) term should account for that. So, maybe I need to consider the entire equation without approximating.Let me write the equation again:( 20a + b = 2,500,001.5 )So, if I don't make any approximations, I can express ( b = 2,500,001.5 - 20a ). But without another equation, I can't find unique values. Maybe the problem expects me to consider that the ( bn ) term is negligible compared to ( an log n ), so ( b ) is much smaller than ( 20a ). Therefore, ( b ) can be considered as approximately ( 2,500,001.5 - 20a ), but without another condition, I can't find exact values.Wait, maybe I need to consider that the original algorithm's time is ( 5n^2 + 3n ), and the refactored is ( an log n + bn ). For large ( n ), the dominant term is ( 5n^2 ), so the refactored algorithm's dominant term ( an log n ) must be half of that, so ( an log n = frac{5}{2}n^2 ). Therefore, ( a = frac{5}{2} times frac{n}{log n} ). But ( a ) is a constant, so this approach doesn't work because ( a ) would depend on ( n ).Alternatively, maybe the problem expects me to equate the leading coefficients. The original leading term is ( 5n^2 ), and the refactored leading term is ( an log n ). Since the refactored is twice as fast, the leading term should be half of the original's leading term. So, ( an log n = frac{5}{2}n^2 ). Therefore, ( a = frac{5}{2} times frac{n}{log n} ). But again, ( a ) would depend on ( n ), which is not allowed.Wait, maybe I'm overcomplicating this. Let's go back to the equation:( 20a + b = 2,500,001.5 )If I consider that ( a ) and ( b ) are constants, and ( n = 10^6 ), then this equation must hold. But without another equation, I can't solve for both ( a ) and ( b ). So, perhaps the problem expects me to express the relationship between ( a ) and ( b ), rather than finding specific values. But the question says "determine the values of ( a ) and ( b )", implying that they are specific.Alternatively, maybe I need to consider that the original time complexity is ( 5n^2 + 3n ), and the refactored is ( an log n + bn ). The ratio of the times is ( T'(n)/T(n) = 1/2 ). So,( frac{an log n + bn}{5n^2 + 3n} = frac{1}{2} )Multiply both sides by ( 5n^2 + 3n ):( an log n + bn = frac{5n^2 + 3n}{2} )Which is the same equation as before. So, again, I have ( 20a + b = 2,500,001.5 ).Since I can't solve for two variables with one equation, maybe the problem expects me to assume that ( b ) is negligible, so ( b = 0 ), leading to ( a = 125,000 ). Alternatively, if ( b ) is not negligible, then I can't find unique values.Wait, maybe the problem expects me to consider that the ( bn ) term is equal to half of the original's ( 3n ) term. So, ( bn = frac{3}{2}n ), leading to ( b = 1.5 ). Then, plugging back into the equation:( 20a + 1.5 = 2,500,001.5 )So,( 20a = 2,500,000 )Therefore,( a = 125,000 )That seems plausible. So, ( a = 125,000 ) and ( b = 1.5 ).But wait, does that make sense? Let's check:Refactored time: ( 125,000 * 10^6 * 20 + 1.5 * 10^6 = 2.5*10^{12} + 1.5*10^6 approx 2.5*10^{12} )Original time: ( 5*10^{12} + 3*10^6 approx 5*10^{12} )So, refactored time is approximately half of the original, which is correct. The ( bn ) term is much smaller than the ( an log n ) term, but it's still part of the equation.Therefore, the values are ( a = 125,000 ) and ( b = 1.5 ).Now, moving on to the second part:The initial space complexity is ( S(n) = cn + d ), and the new space complexity is ( S'(n) = en + f ). The total memory usage reduces by 30% after refactoring for ( n = 10^6 ). So, ( S'(n) = 0.7 S(n) ).So,( en + f = 0.7 (cn + d) )For ( n = 10^6 ):( e*10^6 + f = 0.7 (c*10^6 + d) )Simplify:( 10^6 e + f = 0.7*10^6 c + 0.7d )Rearranged:( 10^6 e + f - 0.7*10^6 c - 0.7d = 0 )Factor out ( 10^6 ):( 10^6 (e - 0.7c) + (f - 0.7d) = 0 )Since ( 10^6 ) is a large number, the term ( e - 0.7c ) must be zero to balance the equation, otherwise, the left side would be dominated by the ( 10^6 ) term. Therefore:( e - 0.7c = 0 ) => ( e = 0.7c )And,( f - 0.7d = 0 ) => ( f = 0.7d )So, the relationship between the constants is ( e = 0.7c ) and ( f = 0.7d ).Therefore, the space complexity constants are scaled down by 70% each.So, summarizing:1. ( a = 125,000 ) and ( b = 1.5 )2. ( e = 0.7c ) and ( f = 0.7d )</think>