Calculus Math Help

Question

A box is to be constructed from a sheet of cardboard that is 10 cm by 60 cm by cutting out squares of length x by x from each corner and bending up the sides.

What is the maximum volume this box could have? (Round your answer to two decimal places.)

So far I have:

(60-2x)(10-2x)(x)

600-120x-20x+4x^2(x)

4x^3-140x^2+600x

Answer 1

You are correct so far, except it might be easier to solve the problem for you if you wrote this as a function. The volume is equal to this and the volume is a function of the variable "x" so:

V(x) = 4x³ - 140x² + 600x

Now, the question is asking you to maximize the volume. To do that, you can take the derivative and set it equal to zero (hopefully you've discussed that).

So V'(x) = 12x² - 280x + 600

and to find the maximum: 12x² - 280x + 600 = 0 and solve:

3x² - 70x + 150 = 0

Using the quadratic formula: x = 2.39 and 20.95

Obviously x cannot be 20.95 because the cardboard is only 10 cm along one side. So the answer is when x = 2.39 cm (actual answer to 6 decimals is 2.387059). Plugging that into the volume function we get V(2.387059) = 4(2.387059)³ - 140(2.387059)² + 600(2.387059) = 688.91 cubic cm

Please note that you really need to look at the graph of the function to convince yourself of what is going on. In reality, taking the derivative and setting it equal to zero does not mean you are finding the max, it means you are finding the extremes. But in looking at the graph you can see the volume increases as x goes from zero to 2.39 then starts to decrease as x goes from 2.39 to 20.95. It then increases as x goes from 20.95 to infinity. However, x is limited to being a max of 5 since the cardboard is only 10 cm wide.