Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 12 (a)
• 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 (a) 

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.) (a) Write down an expression for the volume of the box. 
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 

Denote the length and width of the enclosed box by and , respectively. Then, using the given information, write them in terms of the height of the box and the length of the square sheet. 
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. By the Hint, we first denote the length and width of the enclosed box by and , respectively. Its height is given by . Then, the volume of the box is .
.
On the other hand, the height of the square sheet is same with the summation of double of height of the box and double of the width of the box, so that .
Therefore, and can be expressed as and . Plugging these into the volume formula, we get .
