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

Question

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

Theodore Abad · Accepted Answer

The sum of the volume of the prism and the half-sphere.

The prism's volume is: #l*w*h=V# #16*16*18 = 4608# cubed units The upper sphere's volume: #4/3pir^3=V# but it is a half sphere, so #2/3pir^3=V# #pi2/3*8^3 = 1072# cubed units Put them all together. #1072+4608 = 5680# cubed units

Aaliyah Anderson · Answer

undefined To find the volume of the object, we first need to calculate the volume of the prism and the volume of the spherical cap separately, and then add them together.

The volume of the prism can be calculated using the formula:
Volume_prism = base_area_prism * height_prism

The base area of the prism is the area of a square, which can be calculated by squaring the length of one of its sides.
Base_area_prism = side_length^2

Given that the side length of the square-shaped top is equal to the diameter of the cap's base, which is equal to 16 units (twice the cap's radius), we can calculate the base area of the prism:
Base_area_prism = 16^2

Now, we can calculate the volume of the prism:
Volume_prism = Base_area_prism * height_prism

The volume of the spherical cap can be calculated using the formula:
Volume_cap = (1/3) * π * cap_radius^2 * cap_height

Given that the cap's radius is 8 units and its height is 4 units, we can calculate the volume of the cap:
Volume_cap = (1/3) * π * 8^2 * 4

Now, we can find the total volume of the object by adding the volume of the prism and the volume of the cap:
Total_volume = Volume_prism + Volume_cap