A water pipe is a that is 30 inches long, with a of 1 inch. At the end of the cylinder there is a . We want to find the of the water pipe. First, let's picture the situation.
We see that the is a combination of two distinct solids, a hemisphere and a cylinder. Here is what we need to do to find its volume.
- Find the volume of the hemisphere.
- Find the .
- Add the volumes.
Let's do it!
Volume of the Hemisphere
Here we have the hemisphere.
A hemisphere is half of a . We know that the V_s is four thirds the of and the of the .
V_s=4/3Ï€ r^3
Since the volume of a hemisphere is half the volume of a sphere, this is the formula for the volume V_h of a hemisphere.
V_h=1/2 * 4/3Ï€ r^3
From the diagram we can see the radius is 1 inch. We can substitute this value for r in the formula, then we will simplify the right-hand side.
V_h=1/2 * 4/3Ï€ r^3
V_h=1/2 *4/3Ï€ ( 1)^3
V_h=1/2 *4/3Ï€(1)
V_h=1/2 *4/3Ï€
V_h=1*4/2* 3Ï€
V_h=4/6Ï€
V_h=2/3Ï€
V_h = 2/3( 3.14)
V_h=2.093333...
V_h≈ 2.1
The volume of the hemisphere is about 2.1 cubic inches.
Volume of the Cylinder
Here is the cylinder. We want to find its volume V_c.
To do so, we can use the formula for the volume of a cylinder.
V_c=Ï€ r^2 h
From the diagram we can see that the height h of the cylinder is 30 inches and that the radius of the base r is 1 inch. Let's substitute these values into the formula and calculate the volume.
V_c=Ï€ r^2 h
V_c=Ï€ ( 1)^2 ( 30)
V_c=Ï€(1)(30)
V_c=Ï€(30)
V_c=30Ï€
V_c=30( 3.14)
V_c=94.2
The volume of the cylinder is 94.2 cubic inches.
Volume of the Figure
Earlier we noted that the volume of the figure is the sum of the volumes of the hemisphere and the cylinder.
V = V_h+ V_c
Let's substitute for the volumes with the values that we found, and then add them.
V = 2.1+ 94.2 = 96.3
The volume of the figure is about 96.3 cubic inches.
