Related rates, clarification sought

• ktpr2
In summary, the problem asks for the rate at which the diameter decreases when the snowball's surface area decreases at a rate of 1 cm^2/min. To solve, we relate the diameter to the surface area using the equation A = piD^2 and differentiate to get dA/dD = 2piD. Then, we solve for dD/dA by placing the result of dA/dD (2piD) in the differentiation equation, giving us a final answer of 1/(20 cm/min).
ktpr2
Given, "if a snowball melts so that its surface area decreases at a rate of 1 cm^2/min, find the rate at which the diameter decreases when the diameter is 10 cm."

$$\frac{dD}{dt} = \frac{dD}{dA} \frac{dA}{dt}$$ were D is my diameter, A is (surface) area and t is time.

I relate D to A by, $$A = 4pi(\frac{D}{2})^2 = piD^2; A' = 2piD$$ and then solve,

$$\frac{dD}{dt} = 2piD * -1 cm^2/min =$$
$$2pi(10cm) * -1cm^2/min = -20 cm^3/min$$

I know I am wrong because I got volume, not length. When I solve for variables, like A, and differentiate, do i always place result where ever A shows up in the differentiation equation? In this case, dD/dA, so I should write 1/(2piD)?

The book give the answer as $$\frac{1}{20 cm/min}$$

You solved for dA/dD

You want dD/dA

yeah I was suspecting that. thanks.

1. What are related rates?

Related rates is a mathematical concept that involves the study of how the rates of change of two or more related variables are connected. It is commonly used in calculus to solve problems involving changing quantities.

2. How do I identify a related rates problem?

A related rates problem is usually identified by the presence of changing quantities that are related to each other through a known equation or geometric figure. These problems often involve finding the rate of change of one quantity with respect to another.

3. What is the process for solving a related rates problem?

The process for solving a related rates problem involves identifying the changing quantities, determining the relationship between them, differentiating the equation with respect to time, and solving for the desired rate of change. It is important to carefully label and track the units of measurement throughout the problem.

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 and sliding down at a rate of 2 feet per second. At the same time, the bottom of the ladder is moving away from the wall at a rate of 1 foot per second. How fast is the top of the ladder moving down the wall when the bottom of the ladder is 6 feet away from the wall?

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

Some common mistakes to avoid when solving related rates problems include forgetting to differentiate with respect to time, mixing up the variables and their rates of change, and not properly labeling the units of measurement. It is also important to carefully read and understand the problem before attempting to solve it.

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