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