Respuesta :
Answer:
x = 7 ft
h = 7 ft
Step-by-step explanation:
Let call "x " side of the square base and " h " the height of the shed, then
V(s) = x² * h ⇒ 343 = x² * h ⇒ h = 343 / x²
Total cost of the shed is
C(t) = Cost of the base (C₁) + Cost of the roof (C₂ ) + Cost of 4 sides (C₃)
Then
C₁ = 2*x² C₂ = 3*x² and C₃ = 4*2,50*x*h
In C₃ as h = 343 / x² Then C₃ = 10 *343/x C₃ = 3430 /x
Total cost of the shed as a function of x is:
C(x) = 2*x² + 3*x² + 3430 / x (1)
Taking derivatives on both sides of the equation we get:
C´(x) = 10*x - 3430 / x²
C´(x) = 0 ⇒ 10*x - 3430 / x² = 0
x - 343 /x² = 0 ⇒ x³ = 343
x =∛ 343
x = 7 ft
If we plug this value in equation 1 we see C(x) > 0 then for x = 7 , we get a minimun in the function
h = 343 /x²
h = 7 ft
The computation of the equation shows that the dimensions will be 7 feet.
How to solve the equation?
The equation can be illustrated below:
Volume = x²y = 343
Price = 2x² + 3x² + (2.50)(4xy) = 5x² + 10xy.
Therefore, 5x² + 10x(353/x²) = 5x² + 3430/x
Taking the first order derivative, x² = 343
Therefore, x = 7
In conclusion, the dimensions are 7ft.
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