Answer:
--- Radius
--- Height
Explanation:
Given
Object: Can (Cylinder)
![Surface\ Area = 517.8cm^2](https://tex.z-dn.net/?f=Surface%5C%20Area%20%3D%20517.8cm%5E2)
Required
Maximize the volume
The surface area is:
![S.A = 2\pi r^2 + 2\pi rh](https://tex.z-dn.net/?f=S.A%20%3D%202%5Cpi%20r%5E2%20%2B%202%5Cpi%20rh)
Substitute 517.8 for S.A
![517.8 = 2\pi r^2 + 2\pi rh](https://tex.z-dn.net/?f=517.8%20%3D%202%5Cpi%20r%5E2%20%2B%202%5Cpi%20rh)
Divide through by 2
![258.9 = \pi r^2 + \pi rh](https://tex.z-dn.net/?f=258.9%20%3D%20%5Cpi%20r%5E2%20%2B%20%5Cpi%20rh)
Factorize:
![258.9 = \pi r(r + h)](https://tex.z-dn.net/?f=258.9%20%3D%20%5Cpi%20r%28r%20%2B%20h%29)
Divide through by ![\pi r](https://tex.z-dn.net/?f=%5Cpi%20r)
![\frac{258.9}{\pi r} = r + h](https://tex.z-dn.net/?f=%5Cfrac%7B258.9%7D%7B%5Cpi%20r%7D%20%3D%20r%20%2B%20h)
Make h the subject
--- (1)
Volume (V) is calculated as:
![V = \pi r^2h](https://tex.z-dn.net/?f=V%20%3D%20%5Cpi%20r%5E2h)
Substitute (1) for h
![V = \pi r^2(\frac{258.9}{\pi r} - r)](https://tex.z-dn.net/?f=V%20%3D%20%5Cpi%20r%5E2%28%5Cfrac%7B258.9%7D%7B%5Cpi%20r%7D%20-%20r%29)
Open Bracket
![V = 258.9r - \pi r^3](https://tex.z-dn.net/?f=V%20%3D%20258.9r%20-%20%5Cpi%20r%5E3)
Differentiate V
![V' = 258.9 - 3\pi r^2](https://tex.z-dn.net/?f=V%27%20%3D%20258.9%20-%203%5Cpi%20r%5E2)
Set V' to 0
![0 = 258.9 - 3\pi r^2](https://tex.z-dn.net/?f=0%20%3D%20258.9%20-%203%5Cpi%20r%5E2)
Collect Like Terms
![3\pi r^2 = 258.9](https://tex.z-dn.net/?f=3%5Cpi%20r%5E2%20%3D%20258.9)
Divide through by 3
![\pi r^2 = 86.3](https://tex.z-dn.net/?f=%5Cpi%20r%5E2%20%3D%2086.3)
Divide through by ![\pi](https://tex.z-dn.net/?f=%5Cpi)
![r^2 = \frac{86.3}{\pi}](https://tex.z-dn.net/?f=r%5E2%20%3D%20%5Cfrac%7B86.3%7D%7B%5Cpi%7D)
![r^2 = \frac{86.3*7}{22}](https://tex.z-dn.net/?f=r%5E2%20%3D%20%5Cfrac%7B86.3%2A7%7D%7B22%7D)
![r^2 = \frac{604.1}{22}](https://tex.z-dn.net/?f=r%5E2%20%3D%20%5Cfrac%7B604.1%7D%7B22%7D)
Take square root of both sides
![r = \sqrt{\frac{604.1}{22}](https://tex.z-dn.net/?f=r%20%3D%20%5Csqrt%7B%5Cfrac%7B604.1%7D%7B22%7D)
![r = 5.24](https://tex.z-dn.net/?f=r%20%3D%205.24)
Recall that:
![h = \frac{258.9}{\pi r} - r](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B258.9%7D%7B%5Cpi%20r%7D%20-%20r)
Substitute 5.24 for r
![h = \frac{258.9}{\pi * 5.24} - 5.24](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B258.9%7D%7B%5Cpi%20%2A%205.24%7D%20-%205.24)
![h = \frac{258.9*7}{22 * 5.24} - 5.24](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B258.9%2A7%7D%7B22%20%2A%205.24%7D%20-%205.24)
![h = \frac{1812.3}{115.28} - 5.24](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7B1812.3%7D%7B115.28%7D%20-%205.24)
![h = 15.72 - 5.24](https://tex.z-dn.net/?f=h%20%3D%2015.72%20-%205.24)
![h = 10.48](https://tex.z-dn.net/?f=h%20%3D%2010.48)
Hence, the dimension that maximize the volume is:
--- Radius
--- Height