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Overview. Related rates problems involve two (or more) variables that change at the same time, possibly at different rates. If we know how the variables are related, and how fast one of them is changing, then we can figure out how fast the other one is changing.
What is related rate of change?
Related rates of change are simply an application of the chain rule. In related-rate problems, you find the rate at which some quantity is changing by relating it to other quantities for which the rate of change is known.
Why are related rates called related rates?
This is the core of our solution: by relating the quantities (i.e. A and r) we were able to relate their rates (i.e. A′ and r′ ) through differentiation. This is why these problems are called “related rates”!
Related rates come in handy when we have two related quantities and one of their rates of change is much harder to find than the other one. Therefore, the work left with us is just to find the equation that relates the two related quantities, and then use the Chain Rule to differentiate both sides with respect to time.
How do related rates work?
Let’s use our Problem Solving Strategy to answer the question.
- Draw a picture of the physical situation. See the figure.
- Write an equation that relates the quantities of interest. A.
- Take the derivative with respect to time of both sides of your equation. Remember the chain rule.
- Solve for the quantity you’re after.
How are related rates used in engineering?
Related rates help us determine how fast or how slow a certain quantity is changing using the rate of change of the second quantity. Let’s take a look at the example shown below: As time progresses, the water level within the cylinder increases.
Related rate problems generally arise as so-called “word problems.” Whether you are doing assigned homework or you are solving a real problem for your job, you need to understand what is being asked. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.”
How do you respond to related rates?
How are related rates used in real life?
Supposedly, related rates are so important because there are so many “real world” applications of it. Like a snowball melting, a ladder falling, a balloon being blown up, a stone creating a circular ripple in a lake, or two people/boats/planes/animals moving away from each other at a right angle.
Read and understand the problem carefully.
What is a related rate problem?
RELATED RATES . A “related rates” problem is a problem which involves at least two changing quantities and asks you to figure out the rate at which one is changing given sufficient information on all of the others.
What is example for rates?
A rate may be equivalently expressed as an inverse of its value if the ratio of its units are also inverse. For example, 5 miles (mi) per kilowatt-hour (kWh) corresponds to 1/5 kWh/mi (or 200 Wh/mi). Rates are relevant to many aspects of everyday life. For example: How fast are you driving? The speed of car (often expressed in miles per hour) is a rate.