Calculating the Volume of Water in a Cylindrical Tank Lying on Its Side

Imagine a cylindrical tank with a radius of 4 meters and a height of 9 meters. This tank is lying on its side and is filled with water to a depth of 2 meters. How can we determine the volume of water in such a situation? We will explore the method step by step.

Identifying the Parameters

The key parameters of this cylindrical tank are as follows:

Radius (r): 4 meters Length (L): 9 meters Depth of water (h): 2 meters

Step-by-Step Calculation

To find the volume of water, we use the formula for the volume of a segment of a circle combined with the length of the cylinder.

Calculating the Angle θ

The depth of the water forms a segment of the circle. We need to determine the angle θ in radians that subtends the water level.

The formula to find the angle θ is given by:

θ 2 cos-??1 ( r - h r )

Substituting the values:

θ 2 cos-??1 ( 4 - 2 4 ) 2 cos-??1 ( 1 2 ) 2 ? ( π 3 ) 2π 3

Calculating the Area of the Circular Segment

The area of the circular segment can be calculated using the formula:

A r 2 2 ( θ - sin(θ)

Substituting the values:

A 4 2 2 ( 2π 3 - sin ( 2π 3 ) )

Since sin ( 2π 3 ) 3 2 :

A 16 2 ( 2π 3 - 3 2 ) 8 ( 2π 3 - 3 2 )

Calculating the Volume of Water

The volume is given by the area of the segment times the length of the cylinder:

V A ? L 8 ( 2π 3 - 3 2 ) ? 9

Final Calculation:

2π 3 ≈ 2.0944 , 3 2 ≈ 0.8660 Therefore: 2π 3 - 3 2 ≈ 1.2284 Thus: V ≈ 72 ? 1.2284 ≈ 88.429 m^3

Hence, the volume of water in the cylindrical tank is approximately 88.43 cubic meters.

Additional Context for Full Understanding

This problem can be further illustrated by considering variations in the water depth. If the tank is 100% filled to a depth of 8 meters, the volume of water would be:

V 9π8 2^2 144π m^3

If the tank is only half-filled to a depth of 4 meters, the volume of water would be:

V 0.50 ? 144π m^3 72π m^3

And as we have calculated, when the water level is reduced to 2 meters, which is 25% of the 8-meter depth, the volume of water would be:

V 0.25 ? 144π m^3 36π m^3 ≈ 113.097 m^3

These calculations help us understand the volume of water at different depths and provide a practical application of the method used to calculate the volume of the segment of a circle.