Question

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?

Answer 1

`V=(80pi)/3 + 180`

Explanation:

To find the object's volume, we just need to add the volumes of the `color(red)(spherical)` `color(red)(ca p)` and the `color(blue)(prism)`.

The formula for a `color(red)(spherical)` `color(red)(ca p)` is `color(red)(V)=(pih^2)/3(3r-h)`, where `r` is the radius of the sphere, and `h` is the height of the cap.

Let's plug in the values:

`color(red)(V)=(pi(4)^2)/3(3(3)-(4))`

`color(red)V=color(red)((80pi)/3)`

Next, we find the volume of the `color(blue)(prism)`. The volume of a `color(blue)(prism)` is `color(blue)(V)=bh` 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:

`color(blue)(V)=(6)^2(5)`
`color(blue)(V)=(36)(5)`
`color(blue)(V)=color(blue)(180)`

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

`V=color(red)((80pi)/3) + color(blue)(180)`