🧠:<think>First, I need to determine the dimensions of the rectangular room. Given that the length is twice the width and the area is 200 square meters, I can set up the equation (2w^2 = 200), where (w) is the width. Solving for (w), I find that the width is 10 meters and the length is 20 meters.Next, I'll calculate the total number of square tiles needed. Since each tile is 1 square meter, the total number of tiles is equal to the area of the room, which is 200 tiles.Emily wants the number of red tiles to be exactly twice the number of yellow tiles. Let (y) represent the number of yellow tiles. Therefore, the number of red tiles will be (2y). The equation becomes (y + 2y = 200), which simplifies to (3y = 200). Solving for (y), I find that (y) is approximately 66.67. However, since the number of tiles must be a whole number, it's not possible to have a fractional tile. This means that Emily cannot achieve exactly twice as many red tiles as yellow tiles with the given constraints.Moving on to Problem B, Emily observes that the number of English participants is 20% more than the number of Mandarin participants, with a total of 66 participants. Let (m) represent the number of Mandarin participants. Then, the number of English participants is (1.2m). The equation becomes (1.2m + m = 66), which simplifies to (2.2m = 66). Solving for (m), I find that there are 30 Mandarin participants and 36 English participants.To form cross-cultural pairs, each Mandarin participant can pair with an English participant. Since there are 30 Mandarin participants, 30 pairs can be formed. This leaves 6 English participants without a pair.In summary, Emily cannot achieve the exact ratio of red to yellow tiles as desired, and there are 30 Mandarin and 36 English participants, resulting in 30 cross-cultural pairs and 6 participants left without a pair.</think>

🧠:<think>Okay, so I have this problem about a makeup artist creating a new product line inspired by Gigi Williams. The product is a cylindrical container with a conical cap. There are two sub-problems to solve here. Let me tackle them one by one.Starting with Sub-problem 1: I need to calculate the total surface area of the cylindrical container with the conical cap, excluding the base of the container. Hmm, surface area calculations can be a bit tricky, but let me break it down.First, the cylindrical container has a radius of 3 cm and a height of 10 cm. The conical cap has a height that's half of the cylinder's height, so that would be 5 cm, and the radius of the cone is the same as the cylinder's, which is 3 cm.Since we're excluding the base of the container, I don't need to calculate the area of the circular base of the cylinder. However, I do need to consider the lateral surface area of the cylinder and the lateral surface area of the conical cap.Let me recall the formulas for these:1. The lateral surface area (LSA) of a cylinder is given by (2pi r h), where (r) is the radius and (h) is the height.2. The lateral surface area (LSA) of a cone is given by (pi r l), where (l) is the slant height of the cone. To find (l), I can use the Pythagorean theorem since the radius, height, and slant height form a right-angled triangle. So, (l = sqrt{r^2 + h^2}).Alright, let's compute the LSA of the cylinder first.Given:- Radius ((r)) = 3 cm- Height ((h)) = 10 cmSo, LSA of cylinder = (2pi times 3 times 10 = 60pi) cm².Now, moving on to the conical cap.Given:- Radius ((r)) = 3 cm- Height ((h)) = 5 cmFirst, let's find the slant height ((l)) of the cone.(l = sqrt{r^2 + h^2} = sqrt{3^2 + 5^2} = sqrt{9 + 25} = sqrt{34}) cm.So, slant height is (sqrt{34}) cm.Now, LSA of the cone = (pi times 3 times sqrt{34} = 3pisqrt{34}) cm².Therefore, the total surface area of the container with the cap is the sum of the cylinder's LSA and the cone's LSA.Total Surface Area = (60pi + 3pisqrt{34}).Hmm, let me see if I can factor out the (pi) to simplify it.Total Surface Area = (pi (60 + 3sqrt{34})) cm².Is that all? Wait, let me double-check if I missed anything. The problem says to exclude the base of the container, so I didn't include the area of the circular base, which is correct. Also, since the cap is conical, it doesn't have a base either, so I only considered the lateral surfaces. That seems right.Moving on to Sub-problem 2: Calculating the total cost to fill the cylindrical container with foundation, which costs 0.05 per cubic centimeter. The conical cap is hollow, so it doesn't contribute to the volume.Alright, so I need to find the volume of the cylindrical container and then multiply it by the cost per cubic centimeter.The formula for the volume of a cylinder is (V = pi r^2 h).Given:- Radius ((r)) = 3 cm- Height ((h)) = 10 cmSo, Volume = (pi times 3^2 times 10 = pi times 9 times 10 = 90pi) cm³.Now, calculating the cost: 90π cm³ multiplied by 0.05 per cm³.Total Cost = (90pi times 0.05 = 4.5pi) dollars.Wait, let me compute that numerically to make sure. Since π is approximately 3.1416, so 4.5 * 3.1416 ≈ 14.137 dollars. But since the problem says to use π in calculations, I think it's acceptable to leave it in terms of π unless specified otherwise.But let me check the question again. It says, "calculate the total cost to fill the cylindrical container with the foundation." It doesn't specify whether to leave it in terms of π or compute a numerical value. Hmm, the first sub-problem used π, so maybe the second one is expecting the same. So, 4.5π dollars is approximately 14.14, but perhaps we can write it as ( frac{9}{2}pi ) or 4.5π.Wait, 90π * 0.05 is indeed 4.5π. So, that's correct.But let me make sure I didn't make a mistake in the volume calculation. Volume of cylinder is πr²h, so 3 squared is 9, times 10 is 90, so 90π. Yes, that's correct.And the cost is 0.05 per cm³, so 90π * 0.05 = 4.5π. Yep, that seems right.So, summarizing:Sub-problem 1: Total Surface Area = π(60 + 3√34) cm².Sub-problem 2: Total Cost = 4.5π dollars.Wait, but in the first sub-problem, I had 60π + 3π√34. Maybe I can factor out 3π? Let me see:60π + 3π√34 = 3π(20 + √34). That might be a neater way to write it.Alternatively, if I compute the numerical value, it would be approximately:60π ≈ 188.4956 cm²3π√34 ≈ 3 * 3.1416 * 5.8309 ≈ 3 * 3.1416 * 5.8309 ≈ 3 * 18.326 ≈ 54.978 cm²So total surface area ≈ 188.4956 + 54.978 ≈ 243.4736 cm².But since the problem says to use π, I think the exact form is better.Similarly, for the cost, 4.5π is approximately 14.137, but again, since π is specified, 4.5π is fine.Wait, 4.5π is equal to (9/2)π, which is also acceptable.So, to recap:Sub-problem 1: Total Surface Area = 60π + 3π√34 cm², or factored as 3π(20 + √34) cm².Sub-problem 2: Total Cost = 4.5π dollars, or (9/2)π dollars.I think that's all. Let me just verify once more.For the surface area:- Cylinder: 2πrh = 2π*3*10 = 60π- Cone: πrl, where l = √(r² + h²) = √(9 + 25) = √34, so π*3*√34 = 3π√34- Total: 60π + 3π√34, correct.For the volume:- Cylinder: πr²h = π*9*10 = 90π- Cost: 90π * 0.05 = 4.5π, correct.Yes, everything seems to check out.Final AnswerSub-problem 1: The total surface area is boxed{pi (60 + 3sqrt{34})} square centimeters.Sub-problem 2: The total cost to fill the container is boxed{frac{9}{2}pi} dollars.</think>

🧠:<think>First, I'll tackle the first question. The school is purchasing 120 units with a two-tiered discount structure. The first 50 units have a 20% discount, and any units beyond that have a 35% discount. I'll calculate the cost for each tier separately and then sum them up to find the total cost.For the first 50 units:- The standard price per unit is 500.- A 20% discount reduces the price to 400 per unit.- So, the cost for 50 units is 50 * 400 = 20,000.For the remaining 70 units (120 - 50):- The standard price per unit is 500.- A 35% discount reduces the price to 325 per unit.- So, the cost for 70 units is 70 * 325 = 22,750.Adding both amounts gives the total cost: 20,000 + 22,750 = 42,750.Next, for the second question, the supplier has a fixed budget of 40,000. I need to determine the maximum number of units that can be purchased under this budget while maintaining the discount structure.First, I'll calculate how many units can be purchased with the initial 20% discount:- The discounted price per unit is 400.- With 40,000, the number of units is 40,000 / 400 = 100 units.Since 100 units fall within the first tier (up to 50 units), there's no need to consider the second tier in this case. Therefore, the maximum number of units that can be purchased with the 40,000 budget is 100 units.</think>

🧠:<think>First, I need to calculate the number of possible combinations of 5 dishes that can be selected from the 15 available dishes for a single guest. This is a combination problem since the order of dishes doesn't matter. The formula for combinations is C(n, k) = n! / (k!(n - k)!), where n is the total number of dishes and k is the number of dishes to choose.So, C(15, 5) = 15! / (5! * 10!) = 3003 possible combinations.Next, I need to determine if it's possible to serve all 200 guests with unique combinations. Since there are 3003 possible unique combinations and only 200 guests, it is indeed possible to provide each guest with a distinct set of 5 dishes without any repetition.Therefore, the maximum number of guests that can be served with unique tasting menus is 3003, which is more than sufficient to accommodate the 200 guests.</think>