Days 3 & 4 Notes: Related Rates

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Lionel Marsh
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1 AP Calculus Unit 4 Applications of the Derivative Part 1 Days 3 & 4 Notes: Related Rates Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation represents A 4 r Surface Area of a Sphere 2. 3 V 4 3 r Volume of a Sphere 3. a 2 2 c b, where c is is constant V 2 4. r h, where r is constant Volume of a Cylinder 5. x cos 15 V r h 3 Volume of a Cone

2 Example 1: Air is leaking out of an inflated balloon in the shape of a sphere at a rate of 230π cubic centimeters per minute. At the instant when the radius is 4 centimeters, what is the rate of change of the radius of the balloon? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable.

3 Example 2: A stone is dropped into a calm body of water, causing ripples in the form of concentric circles. The radius of the outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined.. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable.

4 Example 3: Water is leaking out of a cylindrical tank at a rate of 3 cubic feet per second. If the radius of the tank is 4 feet, at what rate is the depth of the water changing at any instant during the leak? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined.. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable.

5 Example 4: A cone has a diameter of 10 inches and a height of 15 inches. Water is being poured into the cone so that the height of the water level is changing at a rate of 1.2 inches per second. At the instant when the radius of the expose surface area of the water is 2 inches, at what rate is the volume of the water changing? 1. Identify all of the variables involved in the problem. 2. Identify which, if any, of the variables in the problem that remain constant. 3. Identify the rate(s) that are given and the rate that you wish to find. 4. Write an equation, often a geometric formula or trigonometric equation, that relates all of the variables in the problem for which a rate is given or for which a rate is to be determined.. Substitute any value that represents a variable that is constant throughout the problem. It is important to keep in mind that you can have only one more variable than you have rates. You may have to make a substitution that relates one variable in terms of another. 5. Implicitly differentiate both sides of the equation with respect to time. 6. Substitute all instantaneous rates and values of the variable and solve for the remaining rate or variable.

6 Example 5: A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. a. How fast is the top of the ladder moving down the wall when the base of the ladder is 7 feet from the wall? b. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. c. Find the rate at which the angle formed by the ladder and the wall of the house is changing when the base of the ladder is 9 feet from the wall.

7 Example 6: An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles past the antenna, the rate at which the distance between the antenna and the plane is changing is 240 miles per hour. What is the speed of the plane? Example 7: The radius of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the surface area of the sphere when the radius is 6 inches. Example 8: A spherical balloon is expanding at a rate of 60π cubic inches per second. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches.

8 AP Calculus AB Unit 4 Days 3 & 4 Assignment Name: 1. The radius r and area A of a circle are related by the equation A = πr 2. Write an equation that relates da to dr. dt dt 2. The radius r and surface area S of a sphere are related by the equation S = 4 πr 2. Write an equation that relates ds to dr. dt dt The radius r, height h, and volume V of a right circular cylinder are related by the equation V = πr 2 h. Use this relationship to answer questions How is dv related to dr if h is constant? dt dt 4. How is dv related to dh if r is constant? dt dt 5. How is dv related to dr and dh if neither r nor h is constant? dt dt dt

9 Use the following information to solve problems 6 8. The length of a rectangle is decreasing at the rate of 2 cm/sec while the width is increasing at the rate of 2 cm/sec. When l = 12 cm and w = 5 cm, find each of the rates of change of each quantity indicated below. 6. Area 7. Perimeter 8. Length of a diagonal of the rectangle

10 A spherical balloon is inflated with helium at the rate of 100π ft 3 /min. 9. How fast is the balloon s radius increasing when the radius is 5 feet? 10. How fast is the surface area increasing when the radius is 5 feet? 11. A conical tank with the vertex down is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

11 A ladder is 25 feet long and is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Use this information to complete exercises How fast is the top of the ladder moving down the wall when its base is 15 feet from the wall? 13. Consider the triangle formed by the side of the house, the ladder and the ground. Find the rate at which the area of the triangle is changing when the ladder is 9 feet from the wall. 14. Find the rate at which the angle between the ladder and the ground is changing when the base of the ladder is 7 feet from the wall.

12 Water is flowing at a rate of 50 cubic meters per minute from a concrete conical reservoir. The radius of the reservoir is 45 m and the height is 6 m. Use this information to complete exercises 15 and How fast is the water level falling when the water is 5 meters deep? 16. How fast is the radius of the water s surface changing when the water is 5 meters deep? 17. A man 6 feet tall walks at a rate of 5 feet per second toward a street light that is 16 feet tall. At what rate is the length of his shadow changing when he is 10 feet from the base of the light?

13 18. If the volume of a cube is increasing at a rate of 24 in 3 /min and each edge is increasing at 2 in/min, what is the length of each edge of the cube? 19. If the volume of a cube is increasing at a rate of 24 in 3 /min and the surface area is increasing at 12 in 2 /min, what is the length of each edge of the cube? 20. On a morning when the sun will pass directly overhead, the shadow of an 80 foot building on level ground is 60 feet long. At the moment in question, the angle θ the sun makes with the ground is increasing at the rate of 0.27 radians per minute. At what rate is the length of the shadow decreasing?

Implicit Differentiation

Implicit Differentiation Much of our algebraic study of mathematics has dealt with functions. In pre-calculus, we talked about two different types of equations that relate x and y explicit and implicit.