# How Fast is the Water Level Dropping in the Tank?

• Mitchtwitchita
In summary, the conversation discusses a cylindrical tank leaking water at a rate of 0.2 m3/min. The water level is falling at an unknown rate and the question is how to solve for this rate using the given information. By differentiating the formula for volume with respect to time and using the fact that the radius does not change, the rate of change of the water level can be solved for. The hypothetical scenario also involves solving for volume using given rates of change and initial values of height and radius.
Mitchtwitchita
Water is leaking out of a cylindrical tank , with circular base radius 0.8 m at the rate of 0.2 m3/min. The water level is falling at ...? ..m/min. What is the question mark? (Hint: the radius does not change)

V=(pi)r^2h, dV/dt = 0.2, r = 0.8 ?

Can anyone give me a hand in getting this one started?

Differentiate both sides of V= (pi)r^2 h, remembering that "the radius does not change". Since h is the "water level", "the rate at which the water level is falling" is dh/dt.

So, dV/dt = (pi)r^2*dh/dt
0.2 = (pi)(o.8)^2*dh/dt
dh/dt =0.0995 ?

Hypothetically, how would you solve for volume if the rates of change of the height and radii are given, as well as the radius and height when the derivative would seem to cancel the h in the equation?

## 1. What is a "related rate" in the context of a water leak?

A related rate in the context of a water leak refers to the rate at which the leak is occurring in relation to another variable, such as time or the volume of water leaking.

## 2. How do you calculate the related rate of a water leak?

The related rate of a water leak can be calculated using the formula: dV/dt = A * dh/dt, where dV/dt is the rate of change of volume, A is the cross-sectional area of the leak, and dh/dt is the rate of change of the water's height.

## 3. What factors can affect the related rate of a water leak?

The related rate of a water leak can be affected by various factors, such as the size of the leak, the pressure of the water, the type of material the container is made of, and the temperature of the water.

## 4. How can the related rate of a water leak be used in real-world applications?

The related rate of a water leak can be used in various real-world applications, such as determining the rate at which a container is losing water and how quickly it needs to be refilled, predicting the time it will take for a container to be emptied, and evaluating the efficiency of a water system.

## 5. What are some common strategies for solving related rate problems involving water leaks?

Some common strategies for solving related rate problems involving water leaks include drawing a diagram to visualize the situation, identifying all known and unknown quantities, and setting up an equation using the related rate formula. It is also helpful to use units and keep track of any changes in units as the problem is solved.

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