The volumes of similar solids C and D are 700 ft^3 and 510 ft^3, respectively. The surface area of C is 260 ft^2. What is the surface area of D?

1 Answer
Hammer
Jun 29, 2018

A transformation of a solid by a factor #n# into a similar solid with all sides of lenght #n# times bigger will have its surface area increase by a factor of #n^2# and its volume by #n^3#.

As such, if the factor of similarity of the sides is #m#, meaning that

#P_D/P_C = m#

where #P_C# and #P_D# are the total perimeters of solids #C# and #D#, then:

#A_D/A_C = m^2#

and

#V_D/V_C = m^3#

where #A# and #V# are the areas and volumes, respectively.

#=> A_D/A_C = (V_D/V_C)^(2/3)=> A_D = A_C(V_D/V_C)^(2/3)#

Hence the surface area of #D# is

#A_D = 260*(510/700)^(2/3) = 260 (51/70)^(2/3)~~210.517" ft"^2#

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