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 $10$ $10$ , the cap's height is $7$ $7$ , and the cap's radius is $8$ $8$ . What is the object's volume?

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Answer 1 · 2018-04-20T13:05:36

$\frac{833 π + 7680}{3} cubic units.$ $color(blue)((833pi+7680)/3" cubic units.")$

This is a square prism. The radius of the cap is 8, so the width of the square is 16.

Volume of prism:

Area of base x height.

$16^{2} \times 10 = 2560$ $16^2xx10=2560$

Volume of a spherical cap is given by:

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

$V = \frac{π 7^{2}}{3} (3 (8) - 7)$ $V=(pi7^2)/3(3(8)-7)$

$V = \frac{π 49}{3} (17) = \frac{833 π}{3}$ $V=(pi49)/3(17)=(833pi)/3$

Total volume:

$\frac{833 π}{3} + 2560 = \frac{833 π + 7680}{3}$ $(833pi)/3+2560=(833pi+7680)/3$ 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 10 10 , the cap's height is 77 , and the cap's radius is 88 . What is the object's volume?