How do you find the rate of change of area with respect to radius for a circle?

To find the rate of change of area with respect to radius, differentiate the area formula with respect to time: $$\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$$ .

What is the derivative of the area of a circle with respect to its radius?

The derivative of the area A of a circle with respect to its radius r is: $$\frac{dA}{dr} = 2\text{π}r$$ .

If the radius of a circular pool is increasing at a rate of 3 cm/min, how do you express this mathematically?

This can be expressed as: $$\frac{dr}{dt} = 3 \text{ cm/min}$$ .

How do you calculate the area increase of a circle when the radius is 12 cm and the radius is increasing at 3 cm/min?

Using the chain rule: $$\frac{dA}{dt} = 2\text{π}(12) \times 3 = 72\text{π} \text{ cm}^2/\text{min}$$ .

What is the relationship between the rates of change of volume and radius for a sphere?

The relationship is given by: $$\frac{dV}{dt} = 4\text{π}r^2 \times \frac{dr}{dt}$$ .

What is the relationship between the rates of change of area and radius for a circle?

The relationship is given by: $$\frac{dA}{dt} = 2\text{π}r \times \frac{dr}{dt}$$ .

What is the significance of related rates in real-world applications?

Related rates are significant in various fields such as physics, engineering, and environmental science, where understanding how different quantities change with respect to each other is crucial for problem-solving.

How can you apply related rates to solve problems involving moving objects?

You can apply related rates by identifying the quantities involved, establishing relationships between them, and using derivatives to find the rates of change.

12/11 Related Rates Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

What is a related rate problem?

Back

A related rate problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing. These problems often involve the use of derivatives and the chain rule.

2.

FLASHCARD QUESTION

Front

What is the formula for the volume of a sphere?

Back

3.

FLASHCARD QUESTION

Front

How do you find the rate of change of volume with respect to radius for a sphere?

Back

4.

FLASHCARD QUESTION

Front

What is the derivative of the volume of a sphere with respect to its radius?

Back

5.

FLASHCARD QUESTION

Front

If the radius of a sphere is increasing at a rate of 2 in/sec, how do you express this mathematically?

Back

6.

FLASHCARD QUESTION

Front

How do you calculate the volume increase of a sphere when the radius is 2 in and the radius is increasing at 2 in/sec?

Back

7.

FLASHCARD QUESTION

Front

What is the formula for the area of a circle?

Back

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