Question

Gloria would like to construct a box with volume of exactly 40 ft3 using only metal and wood. The metal costs $13 /ft2 and the wood costs $6 /ft2 . If the wood is to go on the sides, the metal is to go on the top and bottom, and if the length of the base is to be 3 times the width of the base, find the dimensions of the box that will minimize the cost of construction.

Answer 1

The dimensions of the box that will minimize the cost of construction are length = 9 ft, width = 3 ft, and height = 4 ft.

Step-by-step explanation:

To minimize the cost of construction, we need to optimize the surface area of the box since the materials are priced per square foot. Let the width of the box be (w), then the length would be (3w) as per the given condition. The height can be represented as (h). The volume of the box is given by (V = lwh), and since (l = 3w), the volume equation becomes
`\(V = 3w^2h\)`.

Now, we are given that the volume of the box is 40 ft³, so
`3w^2h = 40\)`. To minimize cost, we need to minimize the surface area. The surface area (A) can be expressed as (A = 2lw + 2wh), and substituting (l = 3w) gives
`\(A = 6w^2 + 2wh\)`.

With the volume constraint, we can rewrite the surface area in terms of one variable, (w). Solving the volume equation for \(h\) and substituting it into the surface area equation, we get
`\(A = 6w^2 + (80)/(w)\)`. To minimize (A), we take the derivative of (A) with respect to (w) and set it equal to zero. Solving this equation gives (w = 3), and consequently, (l = 9) and (h = 4).

The optimization of functions with constraints, particularly in the context of minimizing surface area or cost, involves mathematical techniques such as Lagrange multipliers. Understanding these methods can provide deeper insights into problem-solving in various fields.