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?

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Answer 1 · 2018-08-08T10:34:12

#877.69 cu#.
I would review my answer myself and trace bugs, if any.

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#.