So my lecturers gave our class a challange to solve this question a few weeks ago, however he has foegotten about it and I am really keen to know how to solve this quesion as a similar patter may come out for our mid sems. Here is the question:
You work in a company that designs packaging for food products. Your team is tasked to design open top storage bins with square bases. As part of the design, your team has to determine the dimensions of the storage bin that can hold the largest volume given that 192 sq centimeters of material are used to fabricate each bin.
I can easily solve this question by using trial and error or using 3D modeling software to model and solve it. However, my teacher insisted to use calculus to solve this question. I remember him telling us to use maxima and minima to solve this question. Does anyone know how to go around doing this question? Any help is appreciated.
The way I see it:
The constraints of the problem are as follows - i) s^2 + 4hs = 192 (s being base width/length & h being the bin hight) ii) Vol = s^2 * h
Now using (i) u can easily replace h in (ii). you will get:
Vol = s^2 * ( 192 - s^2 ) / 4s
Calculus only comes in at this point. Vol is a function of s, and most likely some sort of curve. What you would want to do is find the maxima of this curve as that would give u Vol-max, which is your required answer.
f(s) = Vol(s) Find f'(s) and f''(s) and solve f'(s) = 0 st. f''(s) < 0.
The value of s would be your required answer. Use that s to find h.