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Answer:
[tex]r = 5.24[/tex] --- Radius
[tex]h = 10.48[/tex] --- Height
Explanation:
Given
Object: Can (Cylinder)
[tex]Surface\ Area = 517.8cm^2[/tex]
Required
Maximize the volume
The surface area is:
[tex]S.A = 2\pi r^2 + 2\pi rh[/tex]
Substitute 517.8 for S.A
[tex]517.8 = 2\pi r^2 + 2\pi rh[/tex]
Divide through by 2
[tex]258.9 = \pi r^2 + \pi rh[/tex]
Factorize:
[tex]258.9 = \pi r(r + h)[/tex]
Divide through by [tex]\pi r[/tex]
[tex]\frac{258.9}{\pi r} = r + h[/tex]
Make h the subject
[tex]h = \frac{258.9}{\pi r} - r[/tex] --- (1)
Volume (V) is calculated as:
[tex]V = \pi r^2h[/tex]
Substitute (1) for h
[tex]V = \pi r^2(\frac{258.9}{\pi r} - r)[/tex]
Open Bracket
[tex]V = 258.9r - \pi r^3[/tex]
Differentiate V
[tex]V' = 258.9 - 3\pi r^2[/tex]
Set V' to 0
[tex]0 = 258.9 - 3\pi r^2[/tex]
Collect Like Terms
[tex]3\pi r^2 = 258.9[/tex]
Divide through by 3
[tex]\pi r^2 = 86.3[/tex]
Divide through by [tex]\pi[/tex]
[tex]r^2 = \frac{86.3}{\pi}[/tex]
[tex]r^2 = \frac{86.3*7}{22}[/tex]
[tex]r^2 = \frac{604.1}{22}[/tex]
Take square root of both sides
[tex]r = \sqrt{\frac{604.1}{22}[/tex]
[tex]r = 5.24[/tex]
Recall that:
[tex]h = \frac{258.9}{\pi r} - r[/tex]
Substitute 5.24 for r
[tex]h = \frac{258.9}{\pi * 5.24} - 5.24[/tex]
[tex]h = \frac{258.9*7}{22 * 5.24} - 5.24[/tex]
[tex]h = \frac{1812.3}{115.28} - 5.24[/tex]
[tex]h = 15.72 - 5.24[/tex]
[tex]h = 10.48[/tex]
Hence, the dimension that maximize the volume is:
[tex]r = 5.24[/tex] --- Radius
[tex]h = 10.48[/tex] --- Height