The surface areas of two similar solids are 340 yd2 and 1,158 yd2. The volume of the larger solid is 1,712 yd3. What is the volume of the smaller solid?

Let us say that,

1 = smaller solid

2 = larger solid

We must remember that the surface area of an object is proportional to the square of their sides, therefore,

SA = k s^2

where k is the constant of proportionality.

By taking the ratio of the 2 similar solids:

SA2 / SA1 = k s2^2 / k s1^2

SA2 / SA1 = s2^2 / s1^2

sqrt (SA2 / SA1) = s2 / s1

s2 / s1 = sqrt (1158 / 340)

While the volume of an object is proportional to the cube of their sides, therefore:

V = k’ s^3

where k’ is the constant of proportionality.

By taking the ratio of the 2 similar solids:

V2 / V1 = k’ s2^3 / k’ s1^3

V2 / V1 = s2^3 / s1^3

1712 / V1 = [sqrt (1158 / 340) ]^3

1712/V1 = 6.28556705

V1 = 272.37 yd^3