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 6 , the cap's height is 4 , and the cap's radius is 6 . What is the object's volume?

Aug 7, 2018

1034.96 cu. I would review,for bugs, if any.

Explanation:

From the dimensions of the spherical cap, the semi-angle that it

subtends at the center of its sphere,

$\alpha$ rad $= \arccos \left(\frac{6 - 4}{6}\right) = \arccos \left(\frac{1}{3}\right) = {70.53}^{o}$

$= 1.23 r a d$,

The side length of the square top of the prism is

$2 \left(\sqrt{{6}^{2} - {2}^{2}}\right) = 8 \sqrt{2}$.

The entire volume = Volume of the cap + volume of the prism

= (( volume of the cone + volume of the

spherical cap ) - volume of the cone) + volume of the prism

$= \left(\frac{4}{3} \left({6}^{3}\right) \alpha \sin \alpha - \frac{1}{3} \pi \left({\left(4 \sqrt{2}\right)}^{2}\right) \left(2\right)\right)$

+ (6)((8sqrt2)^2

= 4/3(6^3) ( 1.23)sqrt( 1 - (1/3)^2 ) - 1/3 pi ((4sqrt2)^2)(2))

+ (6)((8sqrt2)^2

$= 333.98 - 67.02 + 768$

$= 1034.96 c u$