MAT 1234 Calculus I Section 2.8 Related Rates

Next.. WebAssign 2.8. Due Next Monday (Difficulty level *****) to give you more time. Please do not wait until Monday afternoon. (WebAssign 2.9 is also due Monday.) Be sure to do it ASAP. Tutors are available today and tomorrow after class. Write down your solutions carefully!!! One of these type of questions will be on the second exam.

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Preview Define Related Rates How to solve word problems involving Related Rates

Related Rates

Example 1 GO NUTS!

Example 1 GO NUTS!

Example 1 A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 1 Draw a diagram A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 2: Define the variables A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 3: Write down all the information in terms of the variables defined A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 4: Set up a relation between the variables

Step 5: Use differentiation to find the related rate

Expectations In the quizzes and exams, you are expected to include these 5 steps in your solutions.

Example 2 A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

Example 2 A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall? Everyone, try step 1 and 2!

Step 1 Draw a diagram A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

Step 2: Define the variables A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall? Remark: Do not define more variables than necessary.

Step 3: Write down all the information in terms of the variables defined A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

Step 4: Set up a relation between the variables

Step 5: Use differentiation to find the related rate

Review: Similar Triangles Two triangles are similar if and only if one of the following 2 conditions are satisfied 1. Their corresponding angles are the same. 2. The ratio of their corresponding sides are the same.

Review: Similar Triangles In particular: If the corresponding angles are the same, then the ratio of their corresponding sides are the same.

Please wait… We are going to walk through some of the main key points in your classwork. Please do not start your classwork now, not even drawing the diagrams.

Example 3 (Classwork) A street light is mounted at the top of a 12- ft-tall pole. A 6-ft-tall man walks away from the pole with a speed of 4ft/s along a straight path. How fast is the tip of his shadow moving when he is 35 ft from the pole?

Example 3 Wall 12 ft 35 feet 4 ft/s ???? ft/s Man 6 ft

Example 3 12 x 6 z Remark: Do not define more variables than necessary. For example, it is not necessary to define a variable for the length of the shadow.

Example 3 12 x 6 z Remark: Do not define more variables than necessary. For example, it is not necessary to define a variable for the length of the shadow. z-x

Example 3 12 x 6 z z-x

Hint 12 x 6 z z-x Use similar triangles to find a relation between x and z. Solve z in terms of x.

The Answers It turns out that in this problem, the answer is independent of the fact that x=35. This means that the tip of the shadow is moving at a constant rate.