Algebra's Related Rates examples

In summary, algebra's related rates problems have various applications beyond the typical examples found in textbooks. These include rate-time-distance problems with multiple agents, job-type problems involving people or machines, and pipes filling or emptying tanks. These problems can be solved by creating a table with expressions, variables, rates, times, and distances or job quantities. While these types of problems are usually only seen in calculus, they can also be solved algebraically by differentiating static formulas to find rates of change.
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What are some OTHER applications of algebra's related rates problems, beyond the typical examples given as textbook exercises? The textbooks usually emphasize rate-time-distance in which an agent moves at different rates between two situations; or two agents move each at a different rate than the other; and other exercises emphasize two or more agents doing a job, each at a different rate. We can generally create a table to show expressions and variables for the agents, the rates, times, and either distances or job quantities. For the "job" type problems, general examples are people doing a job, or pipes filling or emptying a tank, or a machine/machines performing repetetive tasks.

I am interested to know what other such applications are possible which are typically not used in the algebra textbooks? Anyone know of any examples based on your experiences, or more rare textbook examples?
 
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  • #2
Could you give an example of what you mean by algebraic related rates problems? "Rate" is, by its nature, a calculus problem, not an algebraic problem. The basic idea is that you can take a "static" formula, on that does not depend on time, and by differentiating, convert it to a formula for rates of change.
 
  • #3
Yeah I've only heard of related rates in calc.
 

What is an example of a Related Rates problem in Algebra?

An example of a Related Rates problem in Algebra is a situation where two or more variables are changing simultaneously and the rate of change of one variable is related to the rate of change of another variable. For instance, the height of a cone is changing as water is being poured into it at a constant rate. In this scenario, the volume of water being poured is directly related to the height of the cone.

How do you approach solving a Related Rates problem in Algebra?

The first step in solving a Related Rates problem is to identify the variables involved and the rate at which they are changing. Then, you must determine the equation that relates the variables. Next, you differentiate the equation with respect to time and plug in the given values to solve for the unknown rate of change.

What is the chain rule and why is it important in solving Related Rates problems?

The chain rule is a calculus rule that allows us to find the derivative of a composite function. In Related Rates problems, the rates of change are often expressed in terms of other variables, and the chain rule allows us to find the derivative of these composite functions and solve for the unknown rate of change.

How does the placement of variables in the equation affect the solution of a Related Rates problem?

The placement of variables in the equation is crucial in solving a Related Rates problem. The variables must be placed correctly in the equation in order to accurately represent the relationship between them. If the variables are misplaced, it can lead to an incorrect solution.

Can you provide a real-life example of a Related Rates problem in Algebra?

A real-life example of a Related Rates problem can be seen in the rate at which the length of a tree's shadow changes throughout the day. As the position of the sun changes, the length of the shadow changes as well. The height of the tree and the distance between the tree and the sun are related, and by using the concepts of Related Rates, we can determine the rate at which the length of the shadow changes.

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