A rectangular box is to have a square base and a volume of 4 ft.³. If the material for the base costs 20 cent/square foot, the material for the sides costs 30 cent/square foot, and the material for the top costs 10 cent/square foot, determine the dimensions of the box that can be constructed at minimum cost. B) C) D)

Question

Quizplus · Accepted Answer

Let the side length of the square base be x and the height of the box be h. Then we have:
Volume = base area × height
4 = x²h
h = 4/x²

The cost of the materials is given by:
Cost = base cost + 4(side cost) + top cost

Substituting h into the cost equation, we have:
Cost = 20x² + 4(30xh) + 10x²
Cost = 20x² + 120/x + 10x²

To find the minimum cost, we take the derivative of the cost function and set it equal to zero:
dCost/dx = 40x - 120/x² = 0
x³ = 3
x ≈ 1.44 ft
h = 4/x² ≈ 1.85 ft

Therefore, the dimensions of the box that minimize cost are approximately 1.44 ft by 1.44 ft by 1.85 ft, and the best choice is B.

A Rectangular Box Is to Have a Square Base and a Volume