Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 12 (b)
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 • Q7 • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q9 (c) • Q10 (a) • Q10 (b) • Q10 (c) • Q11 (a) • Q11 (b) • Q11 (c) • Q11 (d) • Q11 (e) • Q12 (a) • Q12 (b) • Q13 (a) • Q13 (b) • Q13 (c) • Q13 (d) •
Question 12 (b) 

Suppose you wish to construct an enclosed box of maximal volume out of an square sheet of cardboard by removing the shaded areas below and folding along the dotted lines. (Each unshaded region will form exactly one side of the box.) (b) Determine the three dimensions (length, width and height) of the box of maximal volume. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Find which maximizes the volume given in part (a). 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. First, we find which maximizes the volume given in part (a) by .
To find its critical point, we take a derivative. Using the product and chain rules, we get .
Therefore, we have if and . In other words, the critical points are and . Plugging these values into , we obtain and Obviously, the maximum is obtained when . To find the corresponding length and width, we recall their relations with in part (a): As a result, we have the maximum volume of the box when the length, width, and height are given by , respectively. 