A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 343 cubic feet. The concrete for the base costs $2 per square foot, the material for the roof costs $3 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed.

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

-2x + 7 = 7

subtract 7 from both sides.

-2x = 0

divide both sides by -2.

x = 0

r = 78539458390583095803458309458039485093485 or just 3

step-by-step explanation: