Calculating Water Level in a New Container Using Volume Calculations

Understanding how to calculate water levels in different containers is a common problem in mathematics and practical applications. This article explains how to determine the water level in a new container after transferring water from one container to another, using a simple volume calculation. We will walk through the steps with a detailed example.

Problem Statement and Relevant Information

A cylindrical tank of water has a height of 10.5 centimeters (cm) and a radius of 20 cm. The water is emptied into a new cylindrical container with a radius of 10 cm. The goal is to find the water level in the new container to two decimal places, using π 22/7.

Step-by-Step Calculation

Step 1: Calculate the Volume of the Original Cylindrical Tank

The formula for the volume V of a cylinder is:

V π r^2 h

For the original tank:

First, calculate the square of the radius:

202 400

Now, substitute back into the volume formula:

V (22/7) times; 400 times; 10.5

Calculate:

400 times; 10.5 4200

So:

V (22/7) times; 4200

Calculate:

22 times; 4200 92400

92400/7 13200

Therefore, the volume of water in the original tank is 13200 cubic centimeters (cm3).

Step 2: Calculate the Height of Water in the New Container

The new container has a radius of 10 cm. Let's denote the height of the water in this container as h. We use the volume formula again for the new container:

V π r^2 h

Substitute the values:

13200 (22/7) times; 10^2 times; h

Calculate:

102 100

So:

13200 (22/7) times; 100 times; h

Calculate:

22 times; 100 2200

2200/7 approx; 314.2857

Solve for h by dividing both sides by 314.2857:

h ≈ 42.00 cm

Conclusion

The level of water in the new container is approximately 42.00 cm.

Additional Considerations

If the height of the first cylinder is not 10.5 cm, the problem cannot be solved accurately. Here are the alternative calculations assuming the height is 20 cm:

Volume of the first tank: π10.52times;20

Let height of the second tank be h.

Volume of the second tank: π102h

π102h π10.52times;20

h 10.5times;10.5times;20/100 22.05 cm

Thus, the level of water in the second container is approximately 22.05 cm.

Key Learnings

Understanding volume calculations in cylinders is critical for problems involving liquid transfer. By mastering these steps, you can determine the water level in a container to any required precision.