Hey Jake - I'm struggling with this question from my maths textbook: "A rectangular box is to have a square base and no top. If it's volume is 500cm^{3}, find the least area of sheet metal of which it can be made." I'm not understanding how all this maximum and minimum stuff is applied to these kind of questions, I thought it just had to do with graphs? Thanks!!

Hey Liiz:

HEY Im jake (obviously not haha). Im a year 12 student who completed my 2u HSC mathematics course in year 11 and lm just happy to help out here. Now, this type of question is amongst one of the most difficult ones in geometrical applications of calculus, and unfortunately in HSC exams there WILL be harder ones. But don't worry, once you have practised enough, you will begin to seize some tricks to approach these questions.

Before I begin answering your question, just a few generally tips to help you answer questions like these where only one number value is provided:

BEFORE YOU DO ANYTHING,

**DRAW A DIAGRAM WITH LABELS**1. Highlight all USEFUL INFORMATIONS

**(in this case, highlight rectangular box, square base, no top, 500cm^3 and least area)**2. Appoint two variables to the unknown sides

**(in this case, I named the side length of the square base as x, and the height of the rectangular box as L)**3. There will be at least one number quantity in every one of these questions in 2u mathematics, so the first equation you should construct, using your name variables to construct an equation that uses the numbers provided by the question

4. Draw out the relationship between the two variables through this equation that you have constructed

5. Construct another equation using your variables and the subject that is asked for in the question

** (In this case, for example, we constructed an Area equation which directly relates to what we are asked to find)**6. Substitute in the equivalent expression of a variable

**(In this case, for example, L = 500/x^2, so we substitute any L we see with 500/x^2)** to reduce the total number of values down to one, so that we can construct an equation entire out of only one variable, which then allows us to perform differentiation

7. Clean up the equation after the substitution to make life easier

8. Differentiate the equation

9. Let this derivative = 0 to find any stationary points (In an Exam,

**YOU MUST STATE "LET dy/dx = 0 TO FIND ANY STATIONARY POINTS, OTHERWISE MARKS MAYBE DEDUCTED!!!**)

10. Solve the derivative equation and find a value for your variable which will be your stationary point

11. Test both sides to show that a local minimum/maximum occurs at your stationary points

12. Substitute your minimum value back into the area equation (

**or maximum value if the question asks for maximum area**) to find the minimum area (

**or the maximum area if you substitute in the maximum value**)

So here is my solution:

Hope you find my solutions clear and useful! If you are confused with anything, dont hesitate to ask!!!

Best Regards

Jacky He