A propane gas tank consists of a cylinder with a hemisphere at each end. Find the volume of the tank if the overall length is 20 feet and the diameter of the cylinder is 6 feet (round to an exact number) (see figure in the website below).
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You have the diameter of the cylinder, and thus the radius of the hemispheres. Find the volume of the sphere formed by the two hemispheres, and add that to the cylinder's volume.
To find the volume of the propane gas tank, we need to calculate the volumes of the cylinder and the two hemispheres and then add them together.
First, let's find the volume of the cylinder:
The diameter of the cylinder is 6 feet, so the radius (r) is half of the diameter, which is 6/2 = 3 feet.
The length of the cylinder is given as 20 feet. Therefore, the height (h) of the cylinder is equal to the length minus twice the radius (2r), which is 20 - 2(3) = 20 - 6 = 14 feet.
The volume (V) of a cylinder can be calculated using the formula:
V = πr^2h
Substituting the values we have:
Vcylinder = π(3^2)(14)
Vcylinder = π(9)(14)
Vcylinder = 126π
Next, let's find the volume of the two hemispheres:
The radius (r) of the hemispheres is also 3 feet.
The volume (V) of a hemisphere can be calculated using the formula:
V = (2πr^3)/3
Substituting the value of the radius, we get:
Vhemisphere = (2π(3^3))/3
Vhemisphere = (2π(27))/3
Vhemisphere = (54π)/3
Vhemisphere = (18π)
Since there are two hemispheres, we need to double the volume of a single hemisphere:
Vhemispheres = 2(18π)
Vhemispheres = 36π
Finally, we can find the total volume of the propane gas tank by adding the volumes of the cylinder and the two hemispheres:
Vtotal = Vcylinder + Vhemispheres
Vtotal = 126π + 36π
Vtotal = 162π
Therefore, the volume of the propane gas tank is 162π cubic feet.
To find the volume of the propane gas tank, we need to break it down into different parts and calculate the volume of each part separately.
Step 1: Calculate the volume of the cylindrical part:
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Given the diameter of the cylinder is 6 feet, the radius (r) is half of the diameter, which is 6/2 = 3 feet.
Since the overall length of the tank is given as 20 feet, the height (h) of the cylinder is 20 - 2r (subtracting the height of the two hemispheres).
Plugging in the values, the volume of the cylindrical part is Vcylinder = π(3)^2(20 - 2(3)) = π(9)(14) = 126π cubic feet.
Step 2: Calculate the volume of each hemisphere:
The formula for the volume of a hemisphere is V = (2/3)πr^3, where r is the radius.
Using the same radius as before (r = 3 feet), the volume of one hemisphere is Vhemisphere = (2/3)π(3)^3 = (2/3)π(27) = 18π cubic feet.
Since there are two hemispheres at each end of the cylinder, the total volume of the hemispheres is 2 * 18π = 36π cubic feet.
Step 3: Calculate the total volume of the tank:
The total volume of the tank is the sum of the volumes of the cylindrical part and the two hemispheres.
Vtotal = Vcylinder + Vhemisphere = 126π + 36π = 162π cubic feet.
Therefore, the volume of the propane gas tank is 162π cubic feet, rounded to an exact number.