answer:1. **Setup**: Let the sides of the triangle be 3x, 4x, and 5x. We observe that 3x < 4x < 5x, which can be labeled as a, b, and c accordingly. 2. **Determine triangle type with Pythagorean Theorem**: - Compute a^2 + b^2 = (3x)^2 + (4x)^2 = 9x^2 + 16x^2 = 25x^2. - Compute c^2 = (5x)^2 = 25x^2. 3. **Comparison**: - Since a^2 + b^2 = c^2, we find that 25x^2 = 25x^2. 4. **Conclusion**: - Due to a^2 + b^2 = c^2, the triangle is a right triangle. Thus, the correct answer is textbf{(A) text{The triangle is a right triangle}}. The final answer is boxed{textbf{(A)}} The triangle is a right triangle

answer:To solve this problem, we start by understanding the relationships between the number of candles Jill made with each scent. First, we know that Jill made 10 almond candles. Given that she had one and a half times as much coconut scent as almond scent, we can calculate the number of coconut candles she made as follows: [10 times 1.5 = 15] This means Jill made 15 coconut candles. Next, since Jill made twice as many lavender candles as coconut candles, we calculate the number of lavender candles as: [15 times 2 = 30] Therefore, Jill made boxed{30} lavender candles.

answer:Given the equation 2a=3b where abneq 0, we aim to find the correct expression among the given options. Starting with the given equation: [2a = 3b] We can manipulate this equation to find the ratio of a to b: [frac{2a}{3b} = frac{3b}{3b}] [ Rightarrow frac{a}{b} = frac{3}{2}] This manipulation directly shows that frac{a}{b} = frac{3}{2}, which corresponds to option C. Therefore, we can immediately conclude that: - Option A, which suggests frac{a}{b} = frac{2}{3}, is incorrect because it contradicts our derived ratio. - Option B, suggesting frac{a}{2} = frac{b}{3}, is a misinterpretation of the given equation and does not directly follow from our manipulation. - Option D, suggesting frac{a}{2} = frac{3}{b}, also does not follow from our manipulation and is thus incorrect. Given the direct derivation from the original equation to frac{a}{b} = frac{3}{2}, we confirm that option C is the correct answer. Therefore, the correct expression that meets the requirements given 2a=3b and abneq 0 is: [boxed{C}]

answer:1. **Set up and solve for the dimensions:** Given the ratio of the length L of the rectangle to its width W is 5:4: [ frac{L}{W} = frac{5}{4} ] Let W = 4k and L = 5k for some constant k. 2. **Total length including semicircles:** The total length given is 50 feet, which includes the length of the rectangle plus the diameters of two semicircles (each having a diameter of W): [ L + W = 50 ] Substituting L = 5k and W = 4k: [ 5k + 4k = 50 quad Rightarrow quad 9k = 50 quad Rightarrow quad k = frac{50}{9} ] Hence, W = 4k = frac{200}{9} feet and L = 5k = frac{250}{9} feet. 3. **Calculate the radius of each semicircle:** The radius r of each semicircle is half of W: [ r = frac{W}{2} = frac{100}{9} text{ feet} ] 4. **Area of the combined semicircles (a full circle):** [ A_{text{circle}} = pi r^2 = pi left(frac{100}{9}right)^2 = frac{10000pi}{81} text{ square feet} ] 5. **Area of the rectangle:** [ A_{text{rectangle}} = L times W = left(frac{250}{9}right) times left(frac{200}{9}right) = frac{50000}{81} text{ square feet} ] 6. **Ratio of the areas:** [ text{Ratio} = frac{A_{text{rectangle}}}{A_{text{circle}}} = frac{frac{50000}{81}}{frac{10000pi}{81}} = frac{50000}{10000pi} = frac{5}{pi} ] 7. **Conclusion:** The ratio of the area of the rectangle to the combined area of the semicircles is frac{5{pi}}. The final answer is boxed{textbf{(A)} frac{5}{pi}}