Answer:
The dimensions that give the minimum surface area are:
Length = 14cm
width = 14cm
height = 14cm
And the minimum surface is:
S = 1,176 cm^2
Step-by-step explanation:
A regular rectangular prism has the measures: length L, width W and height H.
The volume of this prism is:
V = L*W*H
The surface of this prism is:
S = 2*(L*W + H*L + H*W)
If the base of the prism is a square, then we have L = W
Then the equations become:
V = L*L*H = L^2*H
S = 2*(L^2 + 2*H*L)
We know that the volume of the figure is 2744 cm^3
Then:
V = 2744 cm^3 = H*(L^2)
In this equation, we can isolate H.
H = (2744 cm^3)/(L^2)
Now we can replace this on the surface equation:
S = 2*(L^2 + 2*L* (2744 cm^3)/(L^2))
S = 2*L^2 + 4(2744 cm^3)/L
Now we want to minimize the surface area, then we need to find the zeros of the first derivative of S.
S' = 2*(2*L) - 4*(2744 cm^3)/L^2
This is equal to zero when:
0 = 2*(2*L) - 4*(2744 cm^3)/L^2
0 = 4*L*L^2 - 4*(2744 cm^3)
4*(2744 cm^3) = 4*L^3
2744 cm^3 = L^3
∛(2744 cm^3) = L = 14cm
Then the length of the base that minimizes the surface is L = 14.
Then we have:
H = (2744 cm^3)/(L^2) = (2744 cm^3)/(14cm)^2 = 14cm
Then the surface is:
S = 2*(L^2 + 2*L*H) = 2*( (14cm)^2 + 2*(14cm)*(14cm)) = 1,176 cm^2
. The third solution can be easily found using the rational root theorem, and it's equal to -1.