# 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  16 , the cap's height is 8 , and the cap's radius is 5 . What is the object's volume?

Aug 8, 2018

$877.69 c u$.
I would review my answer myself and trace bugs, if any.

#### 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.

Note that the height 8 of the cap is greater than the radius 5 of the

sphere, of which this is a part.

From the dimensions of the opposite spherical cap,

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

its sphere,

$\alpha$ rad 

$= \arccos \left(\frac{8 - 5}{5}\right) = \arccos \left(\frac{3}{5}\right) = \arcsin \left(\frac{4}{5}\right)$

$= {57.13}^{o} =$

$= 0.9273 r a d$.,

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

$2 \left(\sqrt{{5}^{2} - {3}^{2}}\right) = 8$.

The entire volume

V = volume of the whole sphere - volume of the opposite spherical

cap + volume of the rectangular cylinder below

Volume of the opposite 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 = 4/3 pi ( 5^3 ) - ( 4/3 ( 5^3 )( 0.9273 )( 4/5 )

- 1 / 3 pi ( 4^2 )(3)) + (16)(4)^2#

$= 523.59 - \left(148.368 - 50.265\right) + 256$

$= 877.69 c u$.