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Section 3.1 A – Related Rates Consider two quantities related to each other. If one of the quantities changes, so does the other. Suppose we know how fast one of the quantities is changing. Then the question we try to answer: “How fast is the other quantity changing?” Let 𝑄 be some quantity, then is the rate of change of 𝑄 with respect to time. Example 1: Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 60 feet? Picture:

Given:

Question:

Relation:

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Steps to solving related rate problems:

1. Read the problem. Draw a picture if you need to!

2. Write an equation that relates all the quantities.

Sometimes you will have more than one equation.

3. Implicitly differentiate both sides of the equation with respect to time.

4. Substitute values for everything you know into the equation(s) you have.

5. Solve for what you need.

Warning: Do not switch steps 3 and 4! Substitute values after differentiating. Formulas from Geometry you need to know:

1. Circles: 𝐶 = 2휋푟; 𝐴 = 휋푟

2. Rectangles: 𝐴 = 푙푤

3. Triangles: 𝐴 = 𝑏ℎ;

Pythagorean Theorem for right triangles: 𝑎 + 𝑏 = 𝑐

Similar triangles: sides are proportional

4. Spheres: 𝑉 = 휋푟 ; 𝑆𝐴 = 4휋푟

5. Right Circular Cone: 𝑉 = 휋푟 ℎ; 𝐿𝑆𝐴 = 휋푟√푟 + ℎ ; 𝑆𝐴 = 휋푟√푟 + ℎ + 휋푟

6. Right Cylinder: 𝑉 = 휋푟 ℎ; 𝐿𝑆𝐴 = 2휋푟ℎ; 𝑆𝐴 = 2휋푟ℎ + 2휋푟

7. Rectangular Prism: 𝑉 = 푙푤ℎ; 𝑆𝐴 = 2푙푤 + 2푙ℎ + 2푤ℎ

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Example 2: Suppose a spherical balloon is inflated at the rate of 10 cubic centimeters per minute. How fast is the radius increasing when the radius is 5 centimeters? Picture: Given:

Question:

Relation:

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Example 3: A point moves along the curve 푦 = 4푥 + 1 in such a way that the 푦 value is decreasing at the rate of 2 units per second. At what rate is 푥 changing when 푥 = 5?

Picture: Given:

Question:

Relation:

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Example 4: Water is pouring into an inverted cone shaped tank at the rate of 5 cubic feet per minute. The tank is 4 ft. tall and has a radius of 3 ft. How fast is the height of the water rising when it is 2 ft. deep?

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Example 5: A 6-foot man is walking towards a 25 foot lamp post at the rate of 10 ft./sec. How fast is the length of his shadow changing when he is 20 feet from the lamppost? Hint: This example will use similar triangles from Geometry.

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Example 6: A rocket is fired from the ground straight up and a camera located 3000 feet away is following the rocket. If a rocket is rising vertically at the rate of 1000 ft/sec when it is 4000 feet up, how fast is the camera-to-rocket distance changing at the instant?

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Example 7: Using the same conditions for the rocket in the previous example, how fast must the camera elevation angle change at the instant to keep the rocket in sight? 3