# Related Rates Problem: Calculating Sand Leak Rate from Conical Pile Height

• regnar
In summary, the conversation discusses a problem involving a conical pile formed by sand leaking out of a container, with a constant relationship between the altitude and radius of the pile. The problem asks for the rate at which the sand is leaking when the altitude is ten inches, and the conversation provides steps for solving the problem using derivatives and an equation relating volume and the dimensions of the pile.
regnar
Hi, I've tried this too many ways and i can't seem to figure it out. the question is:
As sand leaks out of a hole in a container, it forms a conical pile whose altitude is always the same as its radius. If the height of the pile is increasing at a rate of 6in/min, find the rate at which the sand is leaking out when the altitude is ten inches.

It would be great help, if someone could help me. Thank you

The first step in these type of problems is to identify what you are looking for symbolically. In this case you are trying to solve for the rate at which sand is leaking from the container. So write out what this means in terms of derivatives.

The second step is to find an equation relating what you know to what you are trying to figure out... (Think volume of a cone)

Once you have these pieces, the problem should be fairly straight forward by manipulating your equation to get what you are after in the first step.

Regarding the conical pile, its cross-section is a triangle. Use that triangle to get a relationship between the height of the pile and its diameter (the base of the triangle).

Mark44 said:
Regarding the conical pile, its cross-section is a triangle. Use that triangle to get a relationship between the height of the pile and its diameter (the base of the triangle).

I think the relationship is given (assuming altitude and height are the same quantity). The problem says "it forms a conical pile whose altitude is always the same as its radius."

## 1. What is a related rates problem?

A related rates problem is a type of mathematical problem that involves finding the rate of change of one variable with respect to another variable. This type of problem is commonly used in physics and engineering to model real-world situations.

## 2. How do I solve a related rates problem?

To solve a related rates problem, you first need to identify all of the variables involved and their rates of change. Then, you can use the chain rule from calculus to set up an equation that relates the rates of change of the variables. Finally, you can solve the equation for the desired rate of change.

## 3. What is the chain rule?

The chain rule is a fundamental rule in calculus that allows you to find the derivative of a composite function. In the context of related rates problems, the chain rule is used to find the relationship between the rates of change of different variables.

## 4. Can you give an example of a related rates problem?

One example of a related rates problem is the classic "ladder sliding down a wall" problem. In this problem, you are given the rate at which the base of the ladder is sliding away from the wall and asked to find the rate at which the top of the ladder is sliding down the wall. This problem can be solved using the relationship between the lengths of the ladder, wall, and ground, and their respective rates of change.

## 5. Why are related rates problems important?

Related rates problems are important because they allow us to model and understand real-world phenomena in a mathematical way. They also require a deep understanding of calculus and its applications, making them a valuable tool for students and researchers in science and engineering fields.

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