Question

A rectangular box with a volume of 500 ftcubed is to be constructed with a square base and top. the cost per square foot for the bottom is 15cents​, for the top is 10cents​, and for the sides is 2.5cents. what dimensions will minimize the​ cost?

Answer 1

The volume is given by:
V = hx² = 500
We can then clear the height in terms of x:
h = 500 / x².
Area of ​​the base: x²
Areal from the top: x²
Area of ​​the vertical walls: 4xh = 4x (500 / x²) or 2000 / x
The formula sought will then be:
C = 0.15 (area of ​​base) + 0.10 (area of ​​top) + 0.025 (area of ​​vertical walls)
C = 0.15x² + 0.10x² + 0.025 (2000 / x)
Rewriting:
C = .25x² + 50 / x
We derive and match zero to find the critical point:
0 = 0.5x - 50 / x ^ 2
Clearing x we ​​have:
50 / x ^ 2 = 0.5x
x ^ 3 = 50 / 0.5
x ^ 3 = 100
x = 4.64
We derive again:
C '' (x) = 0.5 + 100 / x ^ 3
We evaluate x = 4.64:
C "(4.64) = 0.5 + 100 / (4.64) ^ 3
C '' (4.64)> 0 (x = 4.64 is a minimum)
Solving
h = 500 / x²
h = 500 / (4.64) ²
gives
h = 23.22.
Answer:
The dimensions of the box are 4.64 x 4.64 x 23.22.
The total cost would be $ 16.16.