Respuesta :
Given:
The sum of the radius and height of the cylinder is 9 units.
To find the largest volume:
Since sum of the radius and height of the cylinder is 9 units.
[tex]\begin{gathered} r+h=9 \\ h=9-r\ldots\ldots\ldots\ldots(1) \end{gathered}[/tex]The formula of volume of the cylinder is,
[tex]\begin{gathered} V=\pi r^2h \\ =\pi\times r^2\times(9-r) \\ V=\pi(9r^2-r^3)\ldots\ldots\ldots\ldots(2) \end{gathered}[/tex]To find the maximum of the function:
Differentiate with respect to r,
[tex]\begin{gathered} \frac{dV}{dr}=0 \\ \pi(18r-3r^2)=0 \\ 18r-3r^2=0 \\ 3r\text{(6-r)=0} \\ r=0,r=6 \end{gathered}[/tex]It is impossible that r=0.
So, let us substitute r=6 in equation (2) we get,
[tex]\begin{gathered} V=\pi(9(6)^2-6^3) \\ =\pi(324-216) \\ =108\pi \\ =108\times\frac{22}{7} \\ \approx339.43\text{ cubic units} \end{gathered}[/tex]Hence, the largest volume of the cylinder is 339.43 cubic units.
