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 length of the top. The prism's height is  9 , the cap's height is 8 , and the cap's radius is 7 . What is the object's volume?

1 Answer
May 26, 2017

$V \approx 3200.76$

Explanation:

We want to do this in two parts, the sphere and the prism.

We know the diameter is equal to the length of the top, and we know that $d = 2 r$

In doing this we determine the diameter to be 14, as the radius is 7.

Now let's start with the prism's volume. The volume of a prism is:

$V = l w h$

Since the top is a square, length and width are the same, 14 (the diameter of the cap). So we plug everything in to get:

$V = {14}^{2} \cdot 9$
$V = 1764$

Now the sphere part. The volume of a sphere is:

$V = \frac{4}{3} \pi {r}^{3}$ But since we are dealing with half a sphere, we will use this:
$V = \frac{2}{3} \pi {r}^{3}$

Plug everything in and we get:

$V = \frac{2}{3} \pi {\left(7\right)}^{3}$
$V = 1372 \frac{\pi}{3}$

Add the two volumes together for your answer:

$1372 \frac{\pi}{3} + 1764 = V$
$V \approx 3200.76$