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 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?

Answer 1

`877.69 cu`.
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

`= 4 / 3 a^3 ( alpha ) sin alpha`,

where `alpha (rad)` 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 ( ( 8 - 5 ) / 5) = arccos ( 3 / 5 )= arcsin ( 4 / 5 )`

`= 57.13^o =`

`= 0.9273 rad`.,

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

`2 (sqrt( 5^2 - 3^2) ) = 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 - (148.368 -50.265 ) + 256`

`=877.69 cu`.