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.