The radius of one of with the maximum volume is 20
The volume of the circular cylinder is the amount of space in the cylinder
The relationship between the radius (r) and height (h) is given as:
[tex]r + h=30[/tex]
Make r the subject
[tex]h = 30 - r[/tex]
The volume of a cylinder is:
[tex]V = \pi r^2 h[/tex]
Substitute [tex]h = 30 - r[/tex]
[tex]V = \pi r^2(30 - r)[/tex]
Expand
[tex]V =30 \pi r^2 -\pi r^3[/tex]
Differentiate the above equation
[tex]V' =60 \pi r -3\pi r^2[/tex]
Set to 0
[tex]60 \pi r -3\pi r^2 = 0[/tex]
Rewrite the above equation as
[tex]3\pi r^2 = 60 \pi r[/tex]
Divide both sides by [tex]\pi r[/tex]
[tex]3r = 60[/tex]
Divide both sides by 3
[tex]r = 20[/tex]
Hence, the radius of one of with the maximum volume is 20
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