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 $4$ $4$ , the cap's height is $2$ $2$ , and the cap's radius is $1$ $1$ . What is the object's volume?

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Answer 1 · 2018-04-21T12:31:51

$\frac{48 + 4 π}{3} cubic units .$ $color(blue)((48+4pi)/3 "cubic units".)$

This is a square prism. The radius of the cap is 1, so the width of the square is 2.

Volume of prism:

Area of base x height.

$2^{2} \times 4 = 16$ $2^2xx4=16$

Volume of a spherical cap is given by:

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

$V = \frac{π 2^{2}}{3} (3 (1) - 2)$ $V=(pi2^2)/3(3(1)-2)$

$V = \frac{4 π}{3} (1) = \frac{4 π}{3}$ $V=(4pi)/3(1)=(4pi)/3$

Total volume:

$16 + \frac{4 π}{3} = \frac{48 + 4 π}{3}$ $16+(4pi)/3=(48+4pi)/3$ 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 length of the top. The prism's height is 4 4 , the cap's height is 22 , and the cap's radius is 11 . What is the object's volume?