# How Do You Solve This Related Rates Problem Involving Two Cars and a Pulley?

• ktpr2
In summary, the correct answer is \frac{10}{\sqrt{133}}. There is something wrong with the relation I construct; first the problem:The diagram they give is similar to an isocelles triangle with PQ running 12 ft down the middle. Point A is on the left, point B is on the right, representing carts A and B respectively.However, when taking the derivative, the equation does not seem to work correctly. I think it may be because I am taking the derivative of a constant instead of a variable.
ktpr2
Im going through odd number related rate problems in preparation for an exam tmmrw. The correct answer is $$\frac{10}{\sqrt{133}}$$. There is something wrong with the relation I construct; first the problem:

Two cars A and B are connected by a rope 39 ft long that passes over a pulley P. The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast is cart B moving towards Q at the instant when cart A is 5 ft from Q?

The diagram they give is similar to an isocelles triangle with PQ running 12 ft down the middle. Point A is on the left, point B is on the right, representing carts A and B respectively.

I figure they want me to find $$\frac{dB}{dt} = \frac{dB}{dA} \frac{dA}{dt}$$ So i expressed point QB in terms of QA and got $$\sqrt{ (39-\sqrt{QA^2+144})^2-144}$$. However this taking the deriviative of this times 2ft/s does not yeild the correct answer. What would be the correct way to relate QB in terms of QA?

You have two right triangles sharing a common leg, with the sum of their hypotenuses constant at 39 feet. You need to work out the relationship between the lengths of the remaining two legs. It appears you know that, but have lost track of something somewhere. You need not solve for B in terms of A. Just take advantage of the fact that AP + BP is constant and find d(BQ)/dt in terms of d(BP)/dt and d(AQ)/dt in terms of d(AP)/dt. How are d(AP)/dt and d(BP)/dt related?

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well, then, I could solve for B in terms of A? Does that just mean that my work above is incorrect? Or that $$\frac{dB}{dt} = \frac{dB}{dA} \frac{dA}{dt}$$ is incorrect? I'm trying to find my conceptual flaw.

Your expression for QB appears to be correct, and the chain rule is correct. Perhaps taking a derivative that is a bit complicated is where you are going wrong, or maybe its finding the length of QB. I have some numbers worked out. What are you getting for QB and for the answer?

ktpr2 said:
well, then, I could solve for B in terms of A? Does that just mean that my work above is incorrect? Or that $$\frac{dB}{dt} = \frac{dB}{dA} \frac{dA}{dt}$$ is incorrect? I'm trying to find my conceptual flaw.

Your expression is right. I get the answer. Just be extra careful when taking the derivative.

ahhh... I made the silly mistake of taking $$\frac{dA}{dB} \frac{dA}{dt}$$, not $$\frac{dB}{dt} = \frac{dB}{dA} \frac{dA}{dt}$$ thank you both for your assistence.

## 1. What is a related rates problem?

A related rates problem is a type of mathematical problem that involves finding the rate at which one quantity changes in relation to another quantity. It is commonly used in calculus to solve real-world problems involving changing variables.

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

The first step in approaching a related rates problem is to carefully read and understand the problem. Then, identify the given information and what needs to be found. Next, draw a diagram and label the variables. From there, use appropriate equations and techniques, such as implicit differentiation, to solve for the unknown rate.

## 3. What are some common mistakes when solving a related rates problem?

Some common mistakes when solving a related rates problem include not identifying all given information, not labeling variables correctly, not properly using calculus techniques, and not checking the units of the final answer.

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

Sure! An example of a related rates problem could be: A ladder is leaning against a wall. The base of the ladder is sliding away from the wall at a rate of 2 feet per second. If the top of the ladder is 10 feet above the ground, how fast is the angle between the ladder and the ground changing when the base of the ladder is 8 feet away from the wall?

## 5. How can I check my solution to a related rates problem?

You can check your solution to a related rates problem by plugging in the known values and the rate of change into the original equation and seeing if it matches the rate of change you calculated. Additionally, you can also check if the units of the final answer make sense in the context of the problem.

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