Related Rates Mini Project

Part 1: Make Quiz Corrections

Part 2: Complete the word problems attached, following the directions.

Part 3: Go on one of the following related rates websites (link on my class website).

Complete 3 problems from either site and check your answers. Complete all work on a separate paper to turn in. Label which problem you are doing.

 http://www.mathscoop.com/calculus/derivatives/applications/relatedrates.php#relatedRatesproblems

 http://www.mathfanatics.com/math/relatedrates/relatedratespractice.php? problem=0

Part 4: Find a youtube video on related rates. Watch it and follow along with the example. Write the URL below.

Part 5: You will be picking one random related rates question to answer in class

(pick a question out of a hat).

Staple everything together, IN ORDER.

Total: 80 points

 60 points from the quiz and word problems attached (each problem from

part 2 will be worth 6 points; this can replace the quiz grade) o If you did well on the quiz and you would rather keep that quiz grade,

you do NOT need to do Part 2. If you choose to do Part 2 anyway for

more practice, you can earn up to 5 extra credit points.

 9 points from the websites (3 points per problem)

 4 points for youtube video example

 7 points for random question in class

Due date: Part 2: Related Rates Word Problems

Steps:

1. Draw a picture and label all the parts that you know.

2. Identify the givens. Determine which givens are constant and which are changing.

3. What are you trying to find?

4. Are there any proportions, ratios, or substitutions that need to be made? Do you need to figure out other missing variables? Does a unit need to be converted? (For example, does the problem say anything about how radius and height are related? Or do you need to turn a diameter into a radius? Or is there a similar triangle with proportions? Or can you figure out the missing side of a triangle with Pythagorean theorem?)

5. Identify the formula(s) you need to use. Make any appropriate substitutions.

6. Derive the formula.

7. Plug in everything you know and solve to find the unknown.

8. Label your answer with appropriate units. (Hint: area should have units squared; volume should have units cubed; theta is usually radians per second)

Common Types of Related Rates Word Problems:

A. Volume B. 2 formulas needed C. Triangle – Pythagorean Theorem D. Triangle – theta angle E. Shadow F. Other/Miscellaneous

Directions: First, read each problem and determine what type of related rates word problem it is. Label as A through F (see above). Then, start off by choosing one of each type of problem to solve. Follow the steps above to help you solve the word problems. When finished, go back and choose any 4 more problems to solve. Helpful formulas:

Volume cone

Volume sphere

2 Surface area sphere: SA = 4πr

1. A conical paper cup 3 inches across the top and 4 inches deep is full of water. The cup springs a leak at the bottom and loses water at the rate of 2 cubic inches per minute. How fast is the water level dropping at the instant when the water is exactly 3 inches deep?

2. Air is being pumped into a spherical balloon at the rate of 7 cubic cm per second. What is the rate of change of the radius at the instant the volume equals 36π?

3. A kite 100 feet above the ground is being blown away from the person holding its string in a direction parallel to the ground at the rate of 10 ft. per second. At what rate must the string be let out when the length of the string already let out is 200 ft?

4. A kite is flying at an angle of elevation of . The kite string is being taken in at the rate of 1 ft. per second. If the angle of elevation does not change, how fast is the kite losing altitude? 5. Sand is dumped off a conveyer belt into a pile at the rate of 2 cubic feet per minute. The sand pile is shaped like a cone whose height and base diameter are always equal. At what rate is the height of the pile growing when the pile is 5 ft. high?

6. A camera is located 50 ft from a straight road along which a car is traveling at 100 feet per second. The camera turns so that it is pointed at the car at all times. In radians per second, how fast is the camera turning as the car passes closest to the camera?

7. A balloon leaves the ground 500 ft away from an observer and rises vertically at the rate of 140 ft per minute. At what rate is the angle of inclination of the observer’s line of sight increasing at the instant when the balloon is exactly 500 ft above the ground?

8. A ladder 13 feet long is leaning against the side of a building. If the foot of the ladder is pulled away from the building at a constant rate of 2 inches per second, how fast is the angle formed by the ladder and the ground changing (in radians per second) at the instant when the top of the ladder is 12 feet above the ground.

9. A ladder 13 feet long is leaning against the side of a building. If the foot of the ladder is pulled away from the building at a constant rate of 8 inches per second, how fast is the area of the triangle formed by the ladder, the building, and the ground changing (in feet per second) at the instant when the top of the ladder is 12 feet above the ground?

3 10. Gas is escaping from a spherical balloon at the rate of 2 ft /min. How fast is the surface area changing when the radius is 12 ft?

11. Sand falling from a chute forms a conical pile whose height is always equal to

4/3 the radius of the base. How fast is the volume changing when the radius of the base is 3 ft and is increasing at the rate of 3 inches/minute. 12. A man 5 feet tall walks at the rate 4 ft/sec directly away from a street light which is 20 feet above the street. At what rate is the length of his shadow changing? Is the length increasing or decreasing?

13. A young child is flying a kite horizontally 120 ft above the ground. The child lets out 2.5 ft of string per second. If we assume that there is no sag in the string, at what speed is the kite moving when there is 130 ft of string out?

14. Oil spilled from a ruptured tanker spreads out in a circle whose area increases 2 at a constant rate of 6 mi /hr. How fast is the radius of the spill increasing when the area is 2 9 mi ?

15. Ship A is currently 15 miles east of P and is moving west at 20 mph. Ship B is currently60 miles south of P and is moving north at 15 mph. At what rate is the distance between them changing after 1 hour? (Hint: Figure out where Ship A and Ship B are after 1 hour and draw a new triangle with this information.) Is the distance increasing or decreasing?

16. The radius of a right circular cylinder is increasing at a rate of 2 in/min and the height is decreasing at a rate of 3 in/min. At what rate is the volume changing when the radius is 8 in and the height is 12 in? Is the volume increasing or decreasing?

17. An aircraft is flying horizontally at a constant height of 4000 ft above a fixed observation point P. At a certain instant, the angle of elevation  is 30° and decreasing, and the speed of the aircraft is 300 mi/hr. How fast is  decreasing at this instant? (Hint: change mph to feet per second)

18. An aircraft is flying horizontally at a constant height of 4000 ft above a fixed observation point P. At a certain instant, the angle of elevation  is 30° and decreasing, and the speed of the aircraft is 300 mi/hr. How fast is the distance between the aircraft and the observation point changing at this instant?