# 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  3 , the cap's height is 7 , and the cap's radius is 8 . What is the object's volume?

Aug 9, 2018

$1693.05 c u$

#### Explanation:

I use the formula:

Volume of the cone-ice like part of a sphere of radius a

$= \frac{4}{3} {a}^{3} \left(\alpha\right) \sin \alpha$,

where $\alpha \left(r a d\right)$ is the semi-vertical angle of the bounding

cone, from the center of the sphere to the periphery of the cap.

From the dimensions of the opposite spherical cap,

the semi-angle that this opposite cap subtends at the center of

its sphere,

Here, $\alpha$ rad 

$= \arccos \left(\frac{8 - 7}{8}\right) = \arccos \left(\frac{1}{8}\right) = \arcsin \left(\frac{\sqrt{63}}{8}\right)$

$= {82.82}^{o} =$

$= 1.4455 r a d$.,

The side length of the square-top of the prism is

$2 \left(\sqrt{{8}^{2} - {1}^{2}}\right) = \sqrt{63}$.

The entire volume

V = volume of the opposite spherical

cap + volume of the rectangular cylinder below

The volume of the spherical cap

= the volume of the con-

ice-like part of the sphere that has this cap as its top - volume of

the cone part. Now,

$V = \frac{4}{3} \left({8}^{3}\right) \left(1.4455\right) \left(\frac{\sqrt{63}}{8}\right)$

- 1 / 3 pi ( (sqrt63)^2 )(1)) + (3)(2sqrt63)^2#

$= 979.05 - 42 + 756$

$= 1693.05 c u$