The level of oil in a storage tank buried in the ground can be found in much the same way as a dipstick is used to determine the oil level in an automobile crankcase. Suppose the ends of the cylindrical storage tank are radius of 3ft and the cylinder is 20ft long. Determine the volume of oil in the tank to the nearest cubic foot if the rod shows a depth of 2ft.
If the tank is on end, then if the depth is d ft, the volume is
π*3^2*d = 9πd ft^3
So, if 2 ft deep, that's 18π=56.5 ft^3
However, if the tank is lying on its side, so the axis is horizontal, then we have to figure the area of a circular segment - the cross-section.
In that case, if the oil is d feet deep where 0<=d<=3, we have the area of oil in the cross-section is
a = 1/2 r^2 (θ-sinθ)
where cos(θ/2) = (r-d)/r
Here, with r=3 and d=2,
θ/2 = 2arccos(1/3) = 2.462
a = 9/2 (2.462-sin(2.462)) = 8.25
So, the volume is 8.25*20 = 165 ft^3
To determine the volume of oil in the cylindrical storage tank, we need to calculate the volume of the entire cylinder and then subtract the volume of the empty space above the indicated oil level.
To find the volume of the cylinder, we can use the formula:
V_cylinder = π * r^2 * h
Where:
- V_cylinder is the volume of the cylinder
- π is a mathematical constant (approximately 3.14159)
- r is the radius of the cylinder (given as 3ft)
- h is the height of the cylinder (given as 20ft)
Plugging in the values:
V_cylinder = 3.14159 * (3ft)^2 * 20ft
V_cylinder ≈ 565.48668 ft^3
Now, to find the volume of the oil in the tank, we need to determine the volume of the empty space above the indicated oil level. This can be calculated using the formula for the volume of a cylinder with a height of 2ft.
V_empty_space = π * r^2 * h
V_empty_space = 3.14159 * (3ft)^2 * 2ft
V_empty_space ≈ 56.54867 ft^3
Finally, we can subtract the volume of the empty space from the volume of the entire cylinder to find the volume of the oil in the tank.
V_oil = V_cylinder - V_empty_space
V_oil ≈ 565.48668 ft^3 - 56.54867 ft^3
V_oil ≈ 508.93801 ft^3
Therefore, the volume of oil in the tank, to the nearest cubic foot, is approximately 509 ft^3.
To determine the volume of oil in the tank, we need to find the volume of the portion of the cylinder that is filled with oil.
First, let's calculate the volume of the entire cylinder. The formula for the volume of a cylinder is given by:
Volume = π * r^2 * h
Where:
π is a constant approximately equal to 3.14159
r is the radius of the cylindrical tank (3ft)
h is the height of the cylindrical tank (20ft)
Volume = π * (3ft)^2 * 20ft
Volume = π * 9ft^2 * 20ft
Volume = π * 180ft^3
Now, let's calculate the volume of the portion of the cylinder that is filled with oil. The formula for the volume of a cylinder segment is given by:
Volume of segment = π * r^2 * h
Where:
r is the radius of the cylindrical tank (3ft)
h is the depth of the oil shown by the rod (2ft)
Volume of segment = π * (3ft)^2 * 2ft
Volume of segment = π * 9ft^2 * 2ft
Volume of segment = π * 18ft^3
To find the volume of oil in the tank, we subtract the volume of the empty portion of the tank from the volume of the entire tank:
Volume of oil = Volume - Volume of segment
Volume of oil = π * 180ft^3 - π * 18ft^3
Volume of oil = π * (180ft^3 - 18ft^3)
Volume of oil ≈ 3.14159 * 162ft^3
Volume of oil ≈ 509.29548ft^3
Therefore, the volume of oil in the tank, to the nearest cubic foot, is approximately 509 cubic feet.