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 #4 #, and the cap's radius is #3 #. 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 lengths of the top.  The prism's height is # 12 #, the cap's height is #4 #, and the cap's radius is #3 #.  What is the object's volume?

Autumn Armstrong · Accepted Answer

#color(blue)("Volume"=(1296+80pi)/3 " cubic units")#

This prism is square in shape; its width is six times the radius of the cap, which is three.

Prism volume:

Base area times height.

#6^2xx12=432#

A spherical cap's volume is determined by:

#V=(pih^2)/3(3r-h)#

#V=(pi(4)^2)/3(3(3)-4)#

#V=(16pi)/3(5)=(80pi)/3#

Total amount:

#432+(80pi)/3=(1296+80pi)/3# cubic units.

Avery Allen · Answer

undefined To find the volume of the object, first calculate the volume of the prism and then the volume of the spherical cap. Then, add these volumes together.

The volume of the prism is given by the formula:
$$ \text{Volume}_{\text{prism}} = \text{base area} \times \text{height} $$

Since the prism has a square base, and its side length is equal to the diameter of the cap, which is $ 2 \times 3 = 6 $ units, the area of the base is $ 6 \times 6 = 36 $ square units.

So, the volume of the prism is:
$$ \text{Volume}_{\text{prism}} = 36 \times 12 = 432 \text{ cubic units} $$

The volume of the spherical cap can be calculated using the formula:
$$ \text{Volume}_{\text{cap}} = \frac{1}{3} \pi h^2 (3r - h) $$

where $ r $ is the radius of the cap and $ h $ is the height of the cap. Substituting the given values:
$$ \text{Volume}_{\text{cap}} = \frac{1}{3} \pi \times 4^2 \times (3 \times 3 - 4) $$
$$ \text{Volume}_{\text{cap}} = \frac{1}{3} \pi \times 16 \times (9 - 4) $$
$$ \text{Volume}_{\text{cap}} = \frac{1}{3} \pi \times 16 \times 5 = \frac{80}{3} \pi $$

Adding the volumes of the prism and the cap:
$$ \text{Total Volume} = 432 + \frac{80}{3} \pi $$

Thus, the volume of the object is $ 432 + \frac{80}{3} \pi $ cubic units.