A rectangular storage container without a lid is being constructed to have a volume of 10m'. The length of its base is twice the width. The material for the base costs $10 per square meter.

Question

A rectangular storage container without a lid is being constructed to have a volume of 10m'. The length of its base is twice the width. The material for the base costs $10 per square meter. The sides costs $6 per square meter. Find the cost of materials for the least expensive such container.

Set up a model for the cost of the material.

Once you have the cost of the material as a function of one variable, use calculus to find the cost of materials for the least expensive such container. Be sure to verify this will be the least expensive container using the first or second derivative test.

Answer 1

Let h = the height of the rectangular box (x and 2x are the width and length of the base).

Then volume V =2x²h. But we know the volume is 10 m³, which allows us to determine an expression for h in terms of x: x = 10/2x² = 5/x².

Now we can develop the cost function. The area of the base is 2x² m² and each square meter costs $10. So an expression for the cost of the base is 20x² (dollars).

The front and the back each have an area of 2xh square meters so a total of 4xh m². Each square meter of the front and back is $6, so 24xh is an expression for the total cost of the front and back depending on the width of the base.

Similarly for the sides the area of one side is xh, so a total of 2xh. Multiplying by $6 gives 12xh dollars.

So the cost function can be modeled by C(x) = 20x² + 36xh. Since we have an expression for h in terms of x we can substitute and end up with the cost function depending on x only.

C(x) = 20x² + 36x(5/x²) {x > 0} (since x represents the length of the base of the box)

C(x) = 20x² + 180x^-1

Now determine when the 1st derivative is equal to zero. Substitute that value into the 2nd derivative to show that that value generates a minimum cost.

Do that work, then check your answer here:

desmos.com/calculator/dubyuchbf8