I have a rectangle that has the perimeter of 38cm. I need to make this rectangle into a baseless cylinder and find the greatest volume of it, by deriving.
so far I came with this: for the rectangle the length is 19 - width
L=(19-w)
W= w
then subbed it into finding the radius of the cylinder:
$r= circumference/2\pi$
$r= w/2\pi$
and finally subbed everything into the volume equation:
$V=\pi r^2 h$
$v= \pi (19-(w/2\pi)) (19 -w)$
but now I'm having trouble to derive it and find the greatest volume.
The width will be the circumference of the base circle, hence its radius will be $\dfrac{w}{2\pi}$ and its area $\dfrac{w^2}{4\pi}$, and the volume of the cylinder: $$V=\smash{\dfrac{w^2}{4\pi }} (19-w)$$ Now$$\frac{\mathrm d\mkern0.5mu V}{\mathrm d\mkern0.5mu w}=\smash{\dfrac1{4\pi}}\bigl(2w(19-w)-w^2\bigr)=\smash{\dfrac{w(38-3w)}{4\pi}}$$ Furthermore, the signs of the derivative on $\mathbf R$ are $$\begin{matrix} w&&0&&\frac{38}3&\\ \hline \frac{\mathrm d\mkern0.5mu V}{\mathrm d\mkern0.5mu w}&-&0&+&0&- \end{matrix}$$ hence there is a local maximum at $\dfrac{38}3$ and this maximum is equal to $$V_\text{max}=\frac1{4\pi}\frac{19^3}{3^3}=\frac{6859}{108\pi}.$$