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?

1 Answer
A. S. Adikesavan
Aug 9, 2018

1693.05 cu

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.

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

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