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?

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Answer 1 · 2018-08-09T12:49:06

# 1693.05 cu#

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.

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 ( ( 8 - 7 ) / 8) = arccos ( 1 / 8 )= arcsin ( sqrt ( 63 )/8 ) #

#= 82.82^o = #

# = 1.4455 rad#.,

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

#2 (sqrt( 8^2 - 1^2) ) = sqrt63#.

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 = 4/3 ( 8^3 )(1.4455 )( sqrt63/8 )#

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

#= 979.05 - 42 + 756#

#= 1693.05 cu#