A rectangular box is to have a square base and a volume of 12 ft3. If the material for the base costs $0.17/ft2, the material for the sides costs $0.10/ft2, and the material for the top costs $0.13/ft2, determine the dimensions (in ft) of the box that can be constructed at minimum cost.

a. length ft

b. width ft

c. height ft

(a)Length =2 feet

(b)Width =2 feet

(c)Height=3 feet

Step-by-step explanation:

Let the dimensions of the box be x, y and z

The rectangular box has a square base.

Therefore, Volume of the box

Volume of the box

The material for the base costs , the material for the sides costs , and the material for the top costs .

Area of the base

Cost of the Base

Area of the sides

Cost of the sides=

Area of the Top

Cost of the Base

Total Cost,

Substituting

To minimize C(x), we solve for the derivative and obtain its critical point

Recall:

Therefore, the dimensions that minimizes the cost of the box are:

(a)Length =2 feet

(b)Width =2 feet

(c)Height=3 feet

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .