Question

A shipping company must design a closed rectangular shipping crate with a square base. The volume is 7680ft3. The material for the top and sides costs $4 per square foot and the material for the bottom costs $11 per square foot. Find the dimensions of the crate that will minimize the total cost of material.

Answer 1

Length = 16 ft

Width = 16 ft

Height = 30 ft

Step-by-step explanation:

We are given the volume;

V = 7680 ft³

Now, let y be the square sides dimension and h be the height of the crate.

Thus means that the volume of the crate will be; V = y × y × h

V = y²h

y²h = 7680

Now, the top is also a square, thus;

Area of top & bottom are each y²

A_top = y²

A_bottom = y²

Now, area of vertical side would be;

A_side = yh

For the four vertical sides, it's;

A_4sides = 4yh

We are told the material for the top and sides costs $4 per square foot

Thus;

Cost of 4 sides is 4(4yh) = 16yh

Cost of top = 4y²

We are told the material for the bottom costs $11 per square foot.

Cost of bottom = 11y²

Total cost is;

T = 16yh + 4y² + 11y²

T = 15y² + 16yh

From earlier volume equation, we saw that;

y²h = 7680

Making h the subject, we have;

h = 7680/y²

Putting this for h in the total cost equation, we have;

T = 15y² + 16y(7680/y²)

T = 15y² + 122880/y

To minimize the cost we need to find the derivative of the total cost and set it equal to zero.

Thus;

dT/dy = 30y - 122880/y²

At dT/dy = 0, we have;

30y - 122880/y² = 0

30y = 122880/y²

Thus;

30y³ = 122880

y³ = 122880/30

y³ = 4096

y = ∛4096

y = 16 ft

Plugging 16 for y in the volume equation, we have;

h = 7680/16²

h = 30 ft

Thus, dimensions are:

Length = 16 ft

Width = 16 ft

Height = 30 ft