# Another Related Rates Problem

• amcavoy
In summary, the conversation discusses a water tank in the shape of an inverted cone with a depth of 10 meters and a top radius of 8 meters. Water is flowing into the tank at a rate of 0.1 cubic meters per minute and leaking out at a rate of 0.001h^2 cubic meters per minute, where h is the depth of the water in meters. The question is whether the tank will ever overflow, and it is determined that the problem can be solved using a separable differential equation. However, it is not necessary to solve it as the maximum height of the water is already known to be 10 meters, which is the height of the cone. Therefore, the tank will not overflow.
amcavoy
A water tank is in the shape of an inverted cone with depth 10 meters and top radius 8 meters. Water is flowing into the tank at 0.1 cubic meters/min but leaking out at a rate of 0.001$$h^2$$ cubic meters/min, where h is the depth of the water in meters. Will the tank ever overflow?

Thoughts:

$$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi (\frac{4}{5}h)^{2}h$$

$$\frac{dV}{dt}=\frac{16}{25}\pi h^{2}\frac{dh}{dt}$$

Now I replace $$\frac{dV}{dt}$$ with 0.1-0.001$$h^2$$. This is where I am stuck. Any suggestions?

Thanks.

This just gives a separable diff equ...

It can be solved but why solve it ? :

If the water reaches a highest point this means that h(t) has a maximum...

dh/dt=0->h=10m...which curiously is exactly the height of the cone

Is it a maximum ? It turns out the second derivative is zero, the third derivative is zero...aso..

So does it overflow ?

Last edited:

To determine if the tank will ever overflow, we need to find the depth of the water when the inflow rate equals the outflow rate. This is when 0.1-0.001h^2 = 0. This can be solved by setting the equation equal to 0 and using the quadratic formula to find the roots. The roots are h=10 and h=-10. Since the depth can't be negative, the only possible solution is h=10 meters. This means that the depth of the water will reach 10 meters, but not exceed it. Therefore, the tank will never overflow.

## 1. What is an "Another Related Rates Problem"?

Another Related Rates Problem is a type of mathematical problem that involves finding the rate of change of one quantity with respect to another quantity. It typically involves multiple variables that are related to each other through an equation, and the goal is to find the rate of change of one variable while holding the others constant.

## 2. How do you approach solving an Another Related Rates Problem?

When solving an Another Related Rates Problem, it is important to first identify all the variables involved and their relationships. Then, use the given information to set up an equation that relates the variables. Next, take the derivative of the equation with respect to time and solve for the desired rate of change. Finally, plug in the values for the other variables to find the numerical answer.

## 3. What are some common real-life applications of Another Related Rates Problems?

Another Related Rates Problems can be found in various fields such as physics, engineering, and economics. Some common real-life applications include finding the rate at which the volume of a balloon changes as it is being inflated, determining the speed of a car based on its distance and time traveled, and calculating the change in surface area of a cone as it is being filled with water.

## 4. What are some tips for solving Another Related Rates Problems more efficiently?

One tip for solving Another Related Rates Problems more efficiently is to carefully label all variables and their units. This will help avoid confusion and ensure that the correct values are being used. It is also helpful to draw a diagram or visualize the problem in order to better understand the relationships between the variables. Additionally, practicing with different types of problems can improve problem-solving skills and speed.

## 5. Are there any common mistakes to avoid when solving Another Related Rates Problems?

One common mistake to avoid when solving Another Related Rates Problems is using the wrong formula or equation. It is important to carefully read and understand the given information in order to correctly set up the problem. Another mistake to avoid is not properly differentiating the equation with respect to time, which can lead to incorrect solutions. Finally, always double-check the units of the final answer to ensure they are consistent with the given information.

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