This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to calculate the ratio of the volumes: Step 1: Understand the relationship between areas, linear dimensions, and volumes of similar figures. For any two similar figures, if the ratio of their corresponding linear dimensions is k, then the ratio of their areas is k^2, and the ratio of their volumes is k^3. Given the areas of two similar figures (even if the question states "triangles," it implies similar 3D figures whose corresponding 2D areas are given), we can find the linear scale factor. Let A_1 and A_2 be the areas, and V_1 and V_2 be the volumes. We are given A_1 = 36 cm^2 and A_2 = 25 cm^2. Step 2: Calculate the ratio of the areas. The ratio of the areas is: (A_1)/(A_2) = (36)/(25) Step 3: Determine the linear scale factor (k). Since the ratio of areas is equal to the square of the linear scale factor (k^2), we have: k^2 = (36)/(25) To find the linear scale factor k, take the square root of both sides: k = sqrt((36)/(25)) = sqrt(36)sqrt(25) = (6)/(5) Step 4: Calculate the ratio of their volumes. The ratio of the volumes of similar figures is equal to the cube of the linear scale factor (k^3): (V_1)/(V_2) = k^3 = ((6)/(5))^3 (V_1)/(V_2) = (6^3)/(5^3) = (216)/(125) The ratio of their volumes is 216:125. The ratio of their volumes is 216:125. 3 done, 2 left today. You're making progress.