Answer:
(8.44 ft. × 8.44 ft × 9.63 ft.)
Step-by-step explanation:
A storage shed has the volume = 686 cubic feet.
686 = x²h ----------------(1)
Let the length of square base = x ft
and height of the shed = h ft
Area of concrete base = x² ft²
Cost to build the base = 8x² [since cost to build the square base = $8 per square feet]
Area of the sides of the shed = 4xh
Cost to build the sides = 4xh (3.50) [since cost to build the sides = $3.50 per square ft]
C = 14xh
Total cost = 8x² + 14xh
(C) = 8x² + (14x) × [tex](\frac{686}{x^2})[/tex] = 8x² + [tex](\frac{9604}{x})[/tex]
To minimize the cost we will take the derivative of c and equate it to 0.
[tex]\frac{dc}{dx}=16x-\frac{9604}{x^2}=0[/tex]
[tex]16x = \frac{9604}{x^2}[/tex]
16x³ = 9604
x³ = [tex]\frac{9604}{16}[/tex]
x³ = 600.25
x = 8.44 ft.
For x = 8.44
686 = (8.44)² h
h = [tex]\frac{686}{(8.44)^2}[/tex]
= 9.63 ft
Therefore, for the minimum cost, dimension of the shed should be (8.44 ft. × 8.44 ft × 9.63 ft.)
