The volumes of two similar solids are 216 ft and 1331 ft. The surface area of the smaller solid is 180 ft. What is the surface area of the larger solid?
180 ft^2 * (1311 / 216)^2
for similar things
area goes as L^2
volume as L^3
volumes of similar shapes are proportional to the cubes of their corresponding sides
so 216 : 1331 = side1^3/side2^3
side1 : side2 = 6 : 11
the areas of similar shapes are proportional to the square of their corresponding sides
180 : x = 6^2 : 11^2
180/x = 36/121
36x = 21780
x = 605
The larger has a surface area of 605 ft^2
Thank you.
To find the surface area of the larger solid, we need to use the concept of ratios and proportions. Since the two solids are similar, their corresponding sides are in proportion to each other.
Let's assume that the volume V1 of the smaller solid is proportional to the volume V2 of the larger solid. This means we can write:
V1/V2 = (side length of smaller solid)^3 / (side length of larger solid)^3
Given that the volumes of the two solids are 216 ft and 1331 ft respectively, we can write the equation as:
216/V2 = (side length of smaller solid)^3 / (side length of larger solid)^3
Simplifying this equation, we get:
(side length of larger solid)^3 = (side length of smaller solid)^3 * V2 / 216
Since we are given the surface area of the smaller solid as 180 ft, we can use the formula for the surface area of a solid to find the side length of the smaller solid.
surface area of smaller solid = 6 * (side length of smaller solid)^2
180 = 6 * (side length of smaller solid)^2
(side length of smaller solid)^2 = 180 / 6
(side length of smaller solid)^2 = 30
(side length of smaller solid) = √30
Now, we can find the side length of the larger solid using the equation we derived earlier:
(side length of larger solid)^3 = (√30)^3 * V2 / 216
Let's solve for the side length of the larger solid:
(side length of larger solid)^3 = 30 * V2 / 216
Since we know V2 is 1331 ft, we can substitute that value in the equation:
(side length of larger solid)^3 = 30 * 1331 / 216
Calculating the right side of the equation, we get:
(side length of larger solid)^3 = 18630 / 216
(side length of larger solid)^3 = 86.25
Taking the cube root of both sides of the equation, we find:
(side length of larger solid) = ∛86.25
Finally, we can find the surface area of the larger solid using the formula for surface area:
surface area of larger solid = 6 * (side length of larger solid)^2
Calculating this, we get:
surface area of larger solid = 6 * (∛86.25)^2
Simplifying further, we find:
surface area of larger solid ≈ 6 * 22.5
surface area of larger solid ≈ 135 ft
Therefore, the surface area of the larger solid is approximately 135 ft.