Word Problem - Applications of differentiation

Question

A rectangular box with a square base and a cover is to have a volume of 10 m3. If the cost of the material for the top and base is $10 per square metre and the cost of the material for the sides is $6 per square metre, find the cost of the cheapest such box.

Answer 1

Let the length, width and height of the box be l,w and h respectively. Since the base is a square, length and width are equal. So the dimensions of the box are w,w, and h.

Now, let's find out constraint. The volume of any box is l * w * h. In our case, the volume equation would be:

V = w * w * h. The volume given is 10. Therefore, w^2*h = 10.

Now let's find a cost function, since the question is asking us to minimize the cost. The general formula would be area * price. Let's look at this piece by piece so that its easier to understand. We will look at the top and bottom first.

The area of the bottom is w * w. The area of the top is identical. So the total area of the top and bottom is 2w^2. The cost for the top and bottom would be, as mentioned before, area * price, which is 2w^2 * 10 = 20w^2.

Let's look at the sides. The area of one side is h * w. Since the two sides are identical, the total area is 2hw. The cost is 6 * 2hw = 12hw.

Finally, let's consider the front and back. The area of the front is also, h * w. So the front and back combined would be 2hw. The cost is 6 * 2hw = 12hw.

The total cost function therefore is C = 20w^2 + 12hw + 12hw. Simplifying, we get 20w^2 + 24hw. We need to take the derivative but we can't do that yet since our function is in terms of two variables. Let's go back to our volume equation.

We had w^2*h =10. Dividing both sides by w^2 we get: h = 10 / (w^2). Let's plug this back into our cost.

We get C = 20w^2 + 24w ( 10 / w^2 ). Simplifying this, we get: C = 20w^2 + 240/w. We can bring the w to the top to get: C = 20w^2 + 240w^-1. Taking the derivative gives us, 40w - 240w^-2. Since we need to find the minimum, we have to set this to 0. This gives use 40w - 240w^-2 = 0. -> 40w = 240w^-2. We can bring the negative exponent down to get 40w = 240 / w^2. Multiplying both sides by w^2 we get 40 w^3 = 240. Dividing by 40 gives w^3 = 6. -> w = cubic root of 6. (or 6^(1/2) )

Plugging this into h gives use h = 10 / 6^(1/2). We know h and w, so let's plug them back into our cost function.

We get C = 20 * [ 6^(1/2) ] ^2 + 24 * [ 10 / 6^(1/2) ] * [ 6^(1/2) ]. Plugging all this into the calculator gives us. C = 306.04

So the cheapest box would be $306.04. Hope I was able to help! Please feel free to reach out with any questions.