In A Tank There Are 800 Lts Of Water. On The Top, A Tube Pours 25 Lts Per Minute Into The Tank, And At The Bottom Of The Tank, By Another Tube, 30 Lts Leave Per Minute. How Many Liters Of Water Will Be In The Tank After 15 Minutes

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Introduction

In this problem, we are given a tank with an initial volume of 800 liters of water. A tube at the top pours 25 liters of water per minute into the tank, while another tube at the bottom removes 30 liters of water per minute. We need to calculate the total amount of water in the tank after 15 minutes.

Understanding the Problem

To solve this problem, we need to understand the concept of net flow rate. The net flow rate is the difference between the rate at which water is added to the tank and the rate at which it is removed. In this case, the net flow rate is 25 liters per minute (added) - 30 liters per minute (removed) = -5 liters per minute.

Calculating the Net Change in Water Level

Since the net flow rate is -5 liters per minute, it means that the tank is losing 5 liters of water per minute. To calculate the total amount of water lost in 15 minutes, we multiply the net flow rate by the time:

-5 liters/minute x 15 minutes = -75 liters

Calculating the Final Water Level

The initial water level in the tank is 800 liters. To find the final water level, we subtract the total amount of water lost from the initial water level:

800 liters - 75 liters = 725 liters

Conclusion

After 15 minutes, the tank will have 725 liters of water. This is because the net flow rate of -5 liters per minute results in a loss of 75 liters of water over the 15-minute period.

Mathematical Formulation

Mathematically, we can represent the problem as follows:

Let V(t) be the volume of water in the tank at time t.

dV/dt = -5 (net flow rate)

V(0) = 800 (initial volume)

To find the final volume, we integrate the differential equation:

∫dV = ∫(-5)dt

V(t) = -5t + C

V(0) = 800

-5(0) + C = 800

C = 800

V(t) = -5t + 800

V(15) = -5(15) + 800

V(15) = 725

Real-World Applications

This problem has real-world applications in various fields, such as:

  • Hydraulics: Calculating the water level in a tank is crucial in hydraulic systems, where the flow rate and pressure are critical factors.
  • Water Treatment: In water treatment plants, the flow rate and water level are essential parameters to ensure efficient treatment and distribution of water.
  • Civil Engineering: Calculating the water level in a tank is necessary in civil engineering projects, such as designing water supply systems and drainage networks.

Conclusion

In conclusion, the problem of calculating the water level in a tank after 15 minutes involves understanding the concept of net flow rate and applying mathematical principles to solve it. The final water level in the tank is 725 liters, which is a result of the net flow rate of -5 liters per minute over the 15-minute period.