Question

A rectangular bin is going to be made with a volume of 492 in3. The base of the bin will be a square and the top will be open. The cost of the material for the base is 0.8 cents per square inch, and the cost of the material for the sides is 0.6 cents per square inch. Determine the dimensions of the bin that will minimize the cost of manufacturing it. What is the minimum cost

Answer 1

base side = 9.037 inches

height = 6.024 inches

Minimum cost = 196 cents

Explanation:

The volume of the bin is given by:

`Volume = side^2 * height`

and the surface area of the bin is given by:

`Surface\ area = side^2 + 4*side*height`

The cost of the bin will be:

`Cost = 0.8*side^2 + 0.6*4*side*height`

`Cost = 0.8*side^2 + 2.4*side*height`

From the volume equation, we have:

`height = 492 / side^2`

Now the cost will be:

`Cost = 0.8*side^2 + 2.4*side*492/side^2`

`Cost = 0.8*side^2 + 1180.8/side`

To find the side that gives the minimum cost, we can find the derivative of Cost in relation to side and then make it equal zero:

Abbreviating Cost as C and side as s, we have:

`dC/ds = 0.8*2*s - 1180.8/s^2`

`1.6s - 1180.8/s^2 = 0`

`1.6s = 1180.8/s^2`

`1.6s^3 = 1180.8`

`s^3 = 738`

`s = 9.037\ in`

Finding the height of the bin, we have:

`height = 492 / 9.037^2`

`height = 6.024\ in`

The minimum cost is:

`Cost = 0.8*9.037^2 + 1180.8/9.037 = 196\ cents`