You are required to construct an open box from a paper by cutting four squared from the corners of the paper. The rectangular piece of paper is 5cm long and 3cm wide. What should be the length of the sides of such four (identical) squares that will create a box of largest volume? What will be the largest volume?

  • 1
    $\begingroup$ Welcome to StackExchange. Please show us what you have done already, as then people are able to help you easier. Also, as a note for the future, please don't just post homework questions like this - people don't take kindly to questions that start 'you are required', 'prove', or 'show' (basically, anything that makes it obvious that it is homework or exam prep) $\endgroup$ – lioness99a May 30 '17 at 7:35
  • 2
    $\begingroup$ It seems a little redundant to say that you need help in the title. That much was understood when you asked a question. (Then again, maybe I'm too nit-picky) $\endgroup$ – infinitylord May 30 '17 at 7:36
  • 1
    $\begingroup$ Please don't delete your question text once it has been answered - leave it there for other people to see if they are stuck on similar questions. You can upvote and accept the answer if it answers your question, and this is seen as a thank you - you and the person who answered will get reputation $\endgroup$ – lioness99a May 30 '17 at 8:14

Let $s$ be the side length of the square which is cut out.

The Volume of the box will be given by $$V = (3-2s)(5-2s)s \space\space\space\space\space\space\text{ (do you see why? Draw a picture to help) }$$ $$\implies V = 4s^3 - 16s^2 + 15s $$

To find extrema, take a derivative and set equal to $0$.

$$\frac{dV}{ds} = 12s^2 - 32s + 15 = 0$$

Use the quadratic formula to find solutions to this equation, and note that $0 < s < 1.5$ (otherwise, you would cut off nothing at all or two full rectangles from the sheet).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.