# A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is 60  and the height of the cylinder is 15 . If the volume of the solid is 70 pi, what is the area of the base of the cylinder?

Aug 26, 2016

$\text{Area of Base of cylinder} = 2 \pi$

#### Explanation:

$\text{The Diagram:}$

color(blue)("We know that the volume of the whole solid is" $70 \pi$

$\left(\mathmr{and}\right)$

color(red)("The sum of the volumes of the cone and cylinder is " $70 \pi$

$\text{Now lets set up an equation for solving the question}$

color(brown)("Volume of cone"=1/3 pir^2h=v_1

color(brown)("Volume of cylinder"=pir^2h=v_2

$\text{(Where "h" is the height and "r" is the radius)}$

$\text{Known values of the variables} :$

color(violet)(h color(violet)("of" color(violet)(v_1=60

color(violet)(h color(violet)("of" color(violet)(v_2=15

$\text{We need to find the value of}$ $\pi {r}^{2}$ $\text{which is the base}$

:.color(orange)(v_1+v_2=70pi

$\rightarrow \frac{1}{3} \pi {r}^{2} h + \pi {r}^{2} h = 70 \pi$

$\rightarrow \frac{1}{3} \pi {r}^{2} 60 + \pi {r}^{2} 15 = 70 \pi$

$\rightarrow \frac{1}{\cancel{3}} ^ 1 \pi {r}^{2} {\cancel{60}}^{20} + \pi {r}^{2} 15 = 70 \pi$

$\rightarrow \pi {r}^{2} 20 + \pi {r}^{2} 15 = 70 \pi$

$\text{Rewrite the equation}$

$\rightarrow 20 \pi {r}^{2} + 15 \pi {r}^{2} = 70 \pi$

$\rightarrow 35 \pi {r}^{2} = 70 \pi$

$\rightarrow \pi {r}^{2} = \frac{{\cancel{70}}^{2} \pi}{\cancel{35}} ^ 1$

rArrcolor(green)(pir^2=2pi