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?

To find the volume of the object, first, we need to calculate the volumes of the prism and the spherical cap, then sum them up.

1. Volume of the prism: [ V_{\text{prism}} = \text{Base area} \times \text{Height} ] The base area of the prism is the square-shaped top, so if its side length is equal to the diameter of the spherical cap, which is 8, then the area is ( 8 \times 8 = 64 ) square units.

[ V_{\text{prism}} = 64 \times 12 = 768 ] cubic units.

2. Volume of the spherical cap: [ V_{\text{cap}} = \frac{1}{3} \pi h^2 (3r - h) ] Given that the height of the cap, ( h = 8 ), and the radius, ( r = 4 ), we can plug in these values:

[ V_{\text{cap}} = \frac{1}{3} \pi (8^2) (3 \times 4 - 8) ] [ V_{\text{cap}} = \frac{1}{3} \pi (64) (12 - 8) ] [ V_{\text{cap}} = \frac{1}{3} \pi (64) (4) ] [ V_{\text{cap}} = \frac{1}{3} \pi (256) ] [ V_{\text{cap}} = \frac{256}{3} \pi ]

Now, sum up the volumes of the prism and the spherical cap to find the total volume of the object:

[ V_{\text{total}} = V_{\text{prism}} + V_{\text{cap}} ] [ V_{\text{total}} = 768 + \frac{256}{3} \pi ]

Thus, the volume of the object is ( 768 + \frac{256}{3} \pi ) cubic units.