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?

1 Answer
Aug 8, 2018

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.