Respuesta :
The maximum volume of the cylinder is 27147.355 at the maximum point [tex]r = \frac{50}{\sqrt{3} }[/tex] .
How do you find the maximum volume of the cylinder?
The formula for the volume of the cylinder v = [tex]\pi[/tex][tex]r^{2}[/tex]h, To find the maximum volume of the cylinder we apply the condition of maxima [tex]\frac{\mathrm{d} v}{\mathrm{d} r}[/tex] = 0.
Let r cm be the radius and h cm be the height of the closed cylinder.
Then, Total surface area of the cylinder = [tex]2\pi r(r+h)[/tex]
[tex]5000[/tex] = [tex]2\pi r^{2} +2\pi rh[/tex]
[tex]h[/tex] = [tex]\frac{5000-2\pi r^{2} }{2\pi rh}[/tex] .............(1)
Volume of the cylinder v = [tex]\pi r^{2} h[/tex]
Substitute the value of [tex]h[/tex] in the above equation
[tex]\Rightarrow[/tex] = [tex]\pi r^{2}[/tex] × [tex]\left [ \frac{5000-2\pi r^{2}}{2\pi r} \right ][/tex]
[tex]\Rightarrow[/tex] = [tex]\frac{r}{2}[/tex] × [tex](5000-2\pi r^{2} )[/tex]
[tex]\Rightarrow[/tex] v = [tex]2500r-\pi r^{3}[/tex] ..............(2)
Now, for the maximum volume of the cylinder [tex]\frac{\mathrm{d} v}{\mathrm{d} r}[/tex] = 0
[tex]\Rightarrow[/tex] [tex]\frac{\mathrm{d} (2500r-\pi r^{3})}{\mathrm{d} r} = 0[/tex]
[tex]\Rightarrow[/tex] [tex]3\pi r^{2} = 2500[/tex]
[tex]\Rightarrow[/tex] [tex]r^{2} = \frac{2500}{3\pi }[/tex]
[tex]\Rightarrow[/tex] [tex]r = \frac{50}{\sqrt{3\pi } }[/tex]
Volume is maximum for [tex]r = \frac{50}{\sqrt{3\pi } }[/tex]
Then, v = [tex]2500r-\pi r^{3}[/tex]
[tex]\Rightarrow[/tex] = [tex]2500 \frac{50}{\sqrt{3\pi } } -\pi (\frac{50}{\sqrt{3\pi } } )^{3}[/tex]
[tex]\Rightarrow[/tex] = [tex]\frac{125000}{\sqrt{3\pi } } - \frac{125000}{3\sqrt{3\pi } }[/tex]
[tex]\Rightarrow[/tex] = [tex]\frac{125000}{\sqrt{3\pi } }[/tex]×[tex]\frac{2}{3}[/tex]
[tex]\Rightarrow[/tex] = [tex]\frac{250000}{3\sqrt{3\pi } }[/tex]
[tex]\Rightarrow[/tex] v = 27147.355
Hence, The maximum volume of the cylinder is 27147.355 at the maximum point [tex]r = \frac{50}{\sqrt{3} }[/tex] .
To learn more about total surface area and volume of the cylinder from the given link
https://brainly.com/question/16095729
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