How Long Does It Take for Water Level to Drop by 30 cm in a Cylindrical Tank? A Comprehensive Analysis

Imagine a cylindrical water tank with a diameter of 140 cm that is filled to the brim. A leak develops at the bottom, causing the tank to lose 33 liters of water every hour. How long will it take for the water level to drop by 30 cm? This article will guide you through the step-by-step mathematical analysis to find the answer.

Understanding the Problem

This problem involves three key components: the dimensions of the cylindrical tank, the rate at which water is lost, and the desired drop in water level. By combining these factors, we can determine the duration required for the water level to fall by a specific height. Let's dive into the solution process.

Step 1: Calculate the Volume of Water Lost Due to a 30 cm Drop

The formula for the volume of a cylinder is:

V πr2h

Where:

r is the radius of the cylinder h is the height or depth of the water.

Given:

Diameter 140 cm, so the radius r 140/2 70 cm Height drop h 30 cm

Now, substituting the values into the volume formula:

V π 702 30

Calculating 702:

702 4900 cm2

Substituting this back into the volume formula:

V π 4900 30 approx; 3.1416 times; 4900 times; 30

V approx; 3.1416 times; 147000 approx; 461815.2 cm3

Converting the volume to liters:

V approx; 461815.2 / 1000 approx; 461.82 liters

Step 2: Determine the Rate of Water Loss

The leak causes 33 liters of water to be lost every hour.

Step 3: Calculate the Time for the Volume Lost to Equal the Volume Corresponding to the 30 cm Drop

To find the time t in hours required for the volume lost to equal the volume corresponding to the 30 cm drop, we can use the formula:

t (Volume to lose) / (Rate of loss)

Substituting the values:

t 461.82 / 33 approx; 14.0 hours

Conclusion

The water level in the tank will fall by 30 cm in approximately 14 hours due to the leak.

Reverse Calculation

Given:

Diameter of cylindrical tank D 140 cm Radius of cylindrical tank r 140/2 70 cm Leaking 33 liter/hour Water level will fall by 30 cm due to a leak in the tank

Let the time taken for the water level to fall by 30 cm be t

Volume calculation for 30 cm height:

V πr2h π 702 30 π 4900 30 147000π cm3

Convert liters to cubic centimeters:

1 liter 1000 cm3, so 33 liters 33000 cm3

Thus, the time to lose 461.82 liters is calculated as:

t (Volume to lose) / (Rate of loss) 461.82 / 33 approx; 14.0 hours

Final Conclusion