Question

A cylindrical frame consisting of 2 circles and 4 vertical support bars are built from a piece of wire 180cm long, cut into 6 sections: 2 circles of one length and 4 straight pieces of another length. Find the lengths of the sections that will maximize the volume of the cylinder. [V_cylinder = πr^2h]

Answer 1

$\text{the length }L=180\text{ of the frame is}$

$L= 2*2\pi{r}+4h$

$\text{where }r\text{ base radius}$

$h\text{ heigth of the frame }$

$180 = 4\pi{r}+4h$

$r = \frac{45-h}{\pi}$

$V = \pi{r^2h}$

$\text{volume increases if } r>1\text{ and } h>1$

$\frac{45-h}{\pi}>1$

$h<45-\pi$

$r= \frac{45-h}{\pi};h\isin(1,45-\pi)$

Answer:$r= \frac{45-h}{\pi};h\isin(1,45-\pi)$