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 $12$ $12$ , the cap's height is $4$ $4$ , and the cap's radius is $3$ $3$ . What is the object's volume?

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Answer 1 · 2018-06-12T16:58:42

$Volume = \frac{1296 + 80 π}{3} cubic units$ $color(blue)("Volume"=(1296+80pi)/3 " cubic units")$

This is a square prism. The radius of the cap is 3, so the width of the square is 6.

Volume of prism:

Area of base x height.

$6^{2} \times 12 = 432$ $6^2xx12=432$

Volume of a spherical cap is given by:

$V = \frac{π h^{2}}{3} (3 r - h)$ $V=(pih^2)/3(3r-h)$

$V = \frac{π {(4)}^{2}}{3} (3 (3) - 4)$ $V=(pi(4)^2)/3(3(3)-4)$

$V = \frac{16 π}{3} (5) = \frac{80 π}{3}$ $V=(16pi)/3(5)=(80pi)/3$

Total volume:

$432 + \frac{80 π}{3} = \frac{1296 + 80 π}{3}$ $432+(80pi)/3=(1296+80pi)/3$ cubic units.

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 12 12 , the cap's height is 44 , and the cap's radius is 33 . What is the object's volume?