# An object is made of a prism with a spherical cap on its square shaped top. The cap's base has a diameter equal to the lengths of the top. The prism's height is 5 , the cap's height is 4 , and the cap's radius is 3 . What is the object's volume?

Jul 29, 2018

$V = \frac{80 \pi}{3} + 180$

#### Explanation:

To find the object's volume, we just need to add the volumes of the $\textcolor{red}{s p h e r i c a l}$ $\textcolor{red}{c a p}$ and the $\textcolor{b l u e}{p r i s m}$.

The formula for a $\textcolor{red}{s p h e r i c a l}$ $\textcolor{red}{c a p}$ is $\textcolor{red}{V} = \frac{\pi {h}^{2}}{3} \left(3 r - h\right)$, where $r$ is the radius of the sphere, and $h$ is the height of the cap.

Let's plug in the values:

$\textcolor{red}{V} = \frac{\pi {\left(4\right)}^{2}}{3} \left(3 \left(3\right) - \left(4\right)\right)$

$\textcolor{red}{V} = \textcolor{red}{\frac{80 \pi}{3}}$

Next, we find the volume of the $\textcolor{b l u e}{p r i s m}$. The volume of a $\textcolor{b l u e}{p r i s m}$ is $\textcolor{b l u e}{V} = b h$ where $b$ is the area of the base of the prism and $h$ is the height of the prism.

Let's plug in the values:

$\textcolor{b l u e}{V} = {\left(6\right)}^{2} \left(5\right)$
$\textcolor{b l u e}{V} = \left(36\right) \left(5\right)$
$\textcolor{b l u e}{V} = \textcolor{b l u e}{180}$

Now, we add the volume of the spherical cap to the prism to get the final object's volume.

$V = \textcolor{red}{\frac{80 \pi}{3}} + \textcolor{b l u e}{180}$