# Am I doing this related rates right?

• MHB
• TheFallen018
In summary, a potter is shaping a cylinder with a pottery wheel, and the height is increasing while the radius is decreasing. The problem is to find the rate at which the radius is changing when the height is increasing at 0.1 cm per second, the radius is 1.5cm, and the length is 7cm. The volume of the cylinder is used to solve this problem, and after differentiating it, the rate of change of the radius is found to be approximately -0.0107 cm/sec. Both parties in the conversation agree that this is the correct answer.
TheFallen018
Hey guys,

I have this related rates problem that I'm working through. I think I might have an answer, but I'm not sure.

Here's the question.
A potter shapes a lump of clay into a cylinder using a pottery wheel.
As it spins, it becomes taller and thinner, so the height, h, is increas-
ing and the radius, r, is decreasing. If the height of the cylinder is
increasing at 0.1 cm per second, find the rate at which the radius is
changing when the radius is 1.5cm and the length is 7cm.

I used the volume of the cylinder for this, and attempted to differentiate it. I ended up with this:

$\frac{d}{dt}V=\frac{d}{dt}(\pi{r}^{2}(t)h(t))$

$0=\pi(2r\frac{dr}{dt}h+\frac{dh}{dt}{r}^{2})$

Which broke down to:

$\frac{dr}{dt}=-\frac{h'{r}^{2}}{2rh}$

Which came out as approximately -0.0107 cm/sec

Does this look about right, or have I gone horribly wrong?

Thanks :)

Looks good. I got the same answer.

I feel an overwhelming need to point out that $$\frac{-h'r^2}{2rh}= \frac{-h'r}{2h}$$.

## 1. What are related rates?

Related rates refer to the study of how the rates of change of two or more quantities are related to each other. This involves finding a derivative of a function with respect to time, where the variables are changing at different rates.

## 2. How do I know if I am solving a related rates problem correctly?

A good way to check if you are solving a related rates problem correctly is to make sure your answer makes sense in the context of the problem. Also, double check your work and make sure you are using the correct formula and units.

## 3. What are some common mistakes to avoid when solving related rates problems?

Some common mistakes to avoid when solving related rates problems include using the wrong formula, not properly identifying the related rates, and not considering all the variables involved in the problem.

## 4. How do I set up a related rates problem?

To set up a related rates problem, start by identifying all the variables involved and their rates of change. Then, determine which variables are related to each other through a given equation. Finally, use the chain rule to find the derivative of the equation with respect to time.

## 5. What are some tips for solving related rates problems efficiently?

Some tips for solving related rates problems efficiently include carefully reading the problem and identifying all the given information, drawing a diagram to visualize the problem, and using the correct formula and units. It is also helpful to break the problem down into smaller steps and to check your answer for reasonableness.

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