# 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  15 , the cap's height is 8 , and the cap's radius is 9 . What is the object's volume?

Nov 6, 2016

Total volume$= \pi \left(- 4 \sqrt[2]{5} - 162 \cdot \arcsin 4 \frac{\sqrt[2]{5}}{9} + 82 \cdot 4 \sqrt[2]{5} - {\left(4 \sqrt[2]{5}\right)}^{3} / 3\right) + 4800$

#### Explanation:

Let' consider a circumference with centre in C(0;-1) and radius 9. Its equation is ${x}^{2} + {\left(y + 1\right)}^{2} = 81$ and its vertical diameter is 8 long in the positive y semiplane (for $y > 0$) just the height of our cap.
The intersect $O I$ of it with the x-axys is the height of the rectangle triangle inscripted inside the vertical semicircunference where the projection of the two catheti are just 8 and 10. As a result, according to the second euclid's theorem $I O = \sqrt[2]{80} = 4 \sqrt[2]{5}$.

Notice that $2 \cdot O I$ is the side of the prism's base, i.e. $L = 8 \sqrt[2]{5}$ whereas its height is $H = 15$, so the prism's volume is
${V}_{p r i s m} = {L}^{2} \cdot H = {\left(8 \cdot \sqrt[2]{5}\right)}^{2} \cdot 15 = 4800$

Moving to the cap's volume, it can be seen as the volume of the solid generated by the arc of circumference in the first quarter of the cartesian axis, whose explicit equation is
$y = \sqrt[2]{81 - {x}^{2}} - 1$ for $0 < x < 4 \sqrt[2]{5}$

It results from the second theorem of Guldino that
${V}_{\text{cap")=2pi*x_("barycentre")*area_("section}}$
${V}_{\text{cap}} = \pi \cdot \frac{{\int}_{0}^{4 \sqrt[2]{5}} {\left(\sqrt[2]{81 - {x}^{2}} - 1\right)}^{2} \mathrm{dx}}{{\int}_{0}^{4 \sqrt[2]{5}} \mathrm{dx}} \cdot {\int}_{0}^{4 \sqrt[2]{5}} \mathrm{dx}$
that can be simplified as
${V}_{\text{cap}} = \pi \cdot {\int}_{0}^{4 \sqrt[2]{5}} \left(- 2 \sqrt[2]{81 - {x}^{2}} + 1 + 81 - {x}^{2}\right) \mathrm{dx}$

Solved the integral we get

${V}_{\text{cap}} = \pi \left(- 2 \cdot \left(\frac{1}{2} x \sqrt[2]{81 - {x}^{2}} + 81 \cdot \arcsin \left(\frac{x}{9}\right)\right) + 82 x - {x}^{3} / 3\right) {|}_{0}^{4 \sqrt[2]{5}}$

${V}_{\text{cap}} = \pi \left(- 4 \sqrt[2]{5} - 162 \cdot \arcsin 4 \frac{\sqrt[2]{5}}{9} + 82 \cdot 4 \sqrt[2]{5} - {\left(4 \sqrt[2]{5}\right)}^{3} / 3\right)$