# The number of square meters in the total surface area of a right circular cylinder, including the top and bottom, is equal to the number of cubic meters in its volume. If the radius of the cylinder is five times its height, what is its volume?

Feb 28, 2018

See a solution process below:

#### Explanation:

We know $r = 5 h$ for the cylinder

The formula for the volume of a cylinder is:

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

We can substitute $5 h$ for $r$ giving:

$V = \pi {\left(5 h\right)}^{2} h$

$V = \pi 25 {h}^{2} h$

$V = \pi 25 {h}^{3}$

$V = 25 \pi {h}^{3}$

The formula for the surface area of a cylinder is:

$A = 2 \pi r h + 2 \pi {r}^{2}$

Again, we can substitute $5 h$ for $r$ giving:

$A = \left(2 \pi 5 h \times h\right) + 2 \pi {\left(5 h\right)}^{2}$

$A = 10 \pi {h}^{2} + 2 \pi 25 {h}^{2}$

$A = 10 \pi {h}^{2} + 50 \pi {h}^{2}$

$A = \left(10 + 50\right) \pi {h}^{2}$

$A = 60 \pi {h}^{2}$

Because the Area is equal to the volume we can equate the two and solve for $h$:

$60 \pi {h}^{2} = 25 \pi {h}^{3}$

$\frac{60 \pi {h}^{2}}{25 \pi {h}^{2}} = \frac{25 \pi {h}^{3}}{25 \pi {h}^{2}}$

$\frac{60 \textcolor{red}{\cancel{\textcolor{b l a c k}{\pi {h}^{2}}}}}{25 \textcolor{red}{\cancel{\textcolor{b l a c k}{\pi {h}^{2}}}}} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{25 \pi}}} {h}^{3}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{25 \pi}}} {h}^{2}}$

$\frac{60}{25} = {h}^{3} / {h}^{2}$

$\frac{5 \times 12}{5 \times 5} = \frac{{h}^{2} \times h}{h} ^ 2$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} \times 12}{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} \times 5} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{h}^{2}}}} \times h}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{h}^{2}}}}}$

$\frac{12}{5} = h$

$h = \frac{12}{5}$

The we can calculate $r$ as $r = 5 \times \frac{12}{5} = 12$

We can substitute these back into the formula for the volume of a cylinder and calculate $V$:

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

$V = \pi \times {12}^{2} \times \frac{12}{5}$

$V = \pi \times 144 \times \frac{12}{5}$

$V = \pi \times \frac{1728}{5}$

$V = \pi \times 345.6$

$V = 345.6 \pi$

Approximating $\pi$ with 3.14 gives:

$V = 345.6 \pi \cong 345.6 \times 3.14 \cong 1085.184$