Question

The total surface area of a closed cylinder is

Answer 1

The maximum volume of the cylinder is 27147.355 at the maximum point  $r = \frac{50}{\sqrt{3} }$ .

How do you find the maximum volume of the cylinder?

The formula for the volume of the cylinder v = $\pi$$r^{2}$h, To find the maximum volume of the cylinder we apply the condition of maxima  $\frac{\mathrm{d} v}{\mathrm{d} r}$ = 0.

Let r cm be the radius and h cm be the height of the closed cylinder.

Then, Total surface area of the cylinder = $2\pi r(r+h)$

                                                        $5000$ = $2\pi r^{2} +2\pi rh$

                                                             $h$ =   $\frac{5000-2\pi r^{2} }{2\pi rh}$ .............(1)

Volume of the cylinder v = $\pi r^{2} h$

Substitute the value of $h$ in the above equation

$\Rightarrow$                                       = $\pi r^{2}$ × $\left [ \frac{5000-2\pi r^{2}}{2\pi r} \right ]$

$\Rightarrow$                                      = $\frac{r}{2}$ ×  $(5000-2\pi r^{2} )$

$\Rightarrow$                                    v = $2500r-\pi r^{3}$ ..............(2)

Now, for the maximum volume of the cylinder  $\frac{\mathrm{d} v}{\mathrm{d} r}$ = 0

$\Rightarrow$                                                            $\frac{\mathrm{d} (2500r-\pi r^{3})}{\mathrm{d} r} = 0$

$\Rightarrow$                                                                      $3\pi r^{2} = 2500$

$\Rightarrow$                                                                          $r^{2} = \frac{2500}{3\pi }$

$\Rightarrow$                                                                          $r = \frac{50}{\sqrt{3\pi } }$

Volume is maximum for  $r = \frac{50}{\sqrt{3\pi } }$  

Then, v = $2500r-\pi r^{3}$

$\Rightarrow$              =  $2500 \frac{50}{\sqrt{3\pi } } -\pi (\frac{50}{\sqrt{3\pi } } )^{3}$                              

$\Rightarrow$             = $\frac{125000}{\sqrt{3\pi } } - \frac{125000}{3\sqrt{3\pi } }$

$\Rightarrow$             = $\frac{125000}{\sqrt{3\pi } }$×$\frac{2}{3}$

$\Rightarrow$             = $\frac{250000}{3\sqrt{3\pi } }$                                                              

$\Rightarrow$          v = 27147.355

Hence, The maximum volume of the cylinder is 27147.355 at the maximum point  $r = \frac{50}{\sqrt{3} }$ .                                                

To learn more about total surface area and volume of the cylinder from the given link

brainly.com/question/16095729

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