Answer:
[tex]Height = 12cm[/tex]
[tex]Radius = 6cm[/tex]
Step-by-step explanation:
Given
Represent volume with v, height with h and radius with r
[tex]V = 432\pi[/tex]
Required
Determine the values of h and r that uses the least amount of material
Volume is calculated as:
[tex]V = \pi r^2h\\[/tex]
Substitute 432π for V
[tex]432\pi = \pi r^2h[/tex]
Divide through by π
[tex]432 = r^2h[/tex]
Make h the subject:
[tex]h = \frac{432}{r^2}[/tex]
Surface Area (A) of a cylinder is calculated as thus:
[tex]A=2\pi rh+2\pi r^2[/tex]
Substitute [tex]\frac{432}{r^2}[/tex] for h in [tex]A=2\pi rh+2\pi r^2[/tex]
[tex]A=2\pi r(\frac{432}{r^2})+2\pi r^2[/tex]
[tex]A=2\pi (\frac{432}{r})+2\pi r^2[/tex]
Factorize:
[tex]A=2\pi (\frac{432}{r} + r^2)[/tex]
To minimize, we have to differentiate both sides and set [tex]A' = 0[/tex]
[tex]A'=2\pi (-\frac{432}{r^2} + 2r)[/tex]
Set [tex]A' = 0[/tex]
[tex]0=2\pi (-\frac{432}{r^2} + 2r)[/tex]
Divide through by [tex]2\pi[/tex]
[tex]0= -\frac{432}{r^2} + 2r[/tex]
[tex]\frac{432}{r^2} = 2r[/tex]
Cross Multiply
[tex]2r * r^2 = 432[/tex]
[tex]2r^3 = 432[/tex]
Divide through by 2
[tex]r^3 = 216[/tex]
Take cube roots of both sides
[tex]r = \sqrt[3]{216}[/tex]
[tex]r = 6[/tex]
Recall that:
[tex]h = \frac{432}{r^2}[/tex]
[tex]h = \frac{432}{6^2}[/tex]
[tex]h = \frac{432}{36}[/tex]
[tex]h = 12[/tex]
Hence, the dimension that requires the least amount of material is when
[tex]Height = 12cm[/tex]
[tex]Radius = 6cm[/tex]
