# 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 , the cap's height is 8 , and the cap's radius is 4 . What is the object's volume?

Aug 11, 2018

color(purple)("Volume of the object " V_o = 512 + (256/3) pi ~~ color(chocolate)(780.08 " cubic units"

#### Explanation:

$\text{Prism's height " H_p = 12, " Radius of sphere " R_c = 4, " Height of cap } {H}_{c} = 8 = {D}_{c} = 2 \cdot {R}_{c}$

It's a sphere over square prism with side of square base and the diameter of sphere same. i.e. ${S}_{p} = {D}_{c} = 8$

$\text{Volume of prism } {V}_{p} = {S}_{p}^{2} \cdot {H}_{p} = 8 \cdot 2 \cdot 12 = 512$

$\text{Volume of sphere } {V}_{s} = \frac{4}{3} \pi {R}_{c}^{3} = \frac{4}{3} \cdot \pi \cdot {4}^{3} = \left(\frac{256}{3}\right) \pi$

$\text{Volume of the object " V_o = V_p + V_s = 512 + (256/3) pi ~~ 780.08 " cubic units}$