The surface areas of two similar figures are given. The volume of the largwe figure is given. Find the volume of the smaller figure if the SA= 25 cm SA= 36cm and the V= 216cm?
I assume SA is surface area and is in cm^2
I assume V is volume and is in cm^3
SA proportional to length squared
Volume proportional to length cubed
length1^2/length2^2 = 25/36
so
length1/length 2 = sqrt(25/36) = .83333
so
length1^3/length2^3 = .83333^3 = .5787
so
V1 = .5787 V2 = 125 cm^3
To find the volume of the smaller figure, we need to set up a ratio of the surface areas of the two figures. Since the two figures are similar, their corresponding side lengths are proportional.
Let's denote the surface area of the smaller figure as SA1, the surface area of the larger figure as SA2, the volume of the smaller figure as V1, and the volume of the larger figure as V2.
We have:
SA1 : SA2 = (side length of the smaller figure)^2 : (side length of the larger figure)^2
In this case, we are given that SA1 = 25 cm², SA2 = 36 cm², and V2 = 216 cm³. Let's substitute these values into the ratio equation:
25 : 36 = (side length of the smaller figure)^2 : (side length of the larger figure)^2
To find the ratio of the side lengths, we take the square root of each side:
√(25 : 36) = √((side length of the smaller figure)^2 : (side length of the larger figure)^2)
Simplifying this equation:
5 : 6 = side length of the smaller figure : side length of the larger figure
Now, let's set up a proportion using the volume ratio of the two figures (V1 : V2) and the surface area ratio (5 : 6).
V1 : V2 = (side length of the smaller figure)^3 : (side length of the larger figure)^3
Substituting the given values:
V1 : 216 = (5/6)^3
To solve for V1, multiply both sides of the equation by 216 and then take the cube root of both sides:
V1 = 216 * (5/6)^3
Calculating the value of V1:
V1 = 216 * (125/216) ≈ 125 cm³
Therefore, the volume of the smaller figure is approximately 125 cm³.