FUNCTION COMPOSITION: CHAINING TOGETHER TWO FUNCTION PROCESSES

Transcription

1 FUNCTION COMPOSITION: CHAINING TOGETHER TWO FUNCTION PROCESSES 104 a. Complete the following table of values showing the number of pounds of rice Noelle purchases for varying number of servings of rice desired. Number of servings of rice desired Amount of rice needed (in pounds) n 105 1

2 b. Define a function f that determines the amount of rice needed as a function of the number of servings of rice desired n. c. The current cost of rice at the Asian market is $0.70 per pound. i. How much will it cost Noelle to purchase 0.8 pounds of rice? 3.4 pounds of rice? 6.2 pounds of rice? $0.56, $2.38, $ c. The current cost of rice at the Asian market is $0.70 per pound. ii. Define a function g that determines the cost of rice in dollars as a function of the amount of rice needed in pounds. g(a) = 0.70a where a is the amount of rice purchased in pounds

3 d. Explain how you can determine the cost of the rice if Noelle wants to buy enough for 7 servings. You need to first determine the number of pounds of rice needed by dividing the number of servings (7) by the number of servings in a pound (2.25). You then need to multiply this amount by 0.70 to determine the cost of the rice for 7 servings, which is $ e. Complete the following table of values relating the number of servings of rice desired and the cost of rice (in dollars). Number of servings of rice cost of rice needed (in dollars)

4 f. Write a formula to determine the cost of n servings of rice. First determine the number of pounds of rice needed for n servings of rice, which is gallons, then multiply this by $0.70 (cost per pound): A pebble is thrown into a lake and the radius of the ripple travels outward at 0.5 meters/second. Your goal is to determine the area inside the ripple for the number of seconds elapsed since the pebble hit the water. Before determining a function to represent this relationship, discuss the following questions in your group. a. Draw a picture of the situation and label the quantities. Imagine how the quantities are changing together

5 a. Draw a picture of the situation and label the quantities. Imagine how the quantities are changing together. The quantities involved are the time since the pebble hit the water, the radius of the circular ripple, and the area enclosed by the circular ripple. 112 b. What quantities are varying (changing) in the situation and how are they changing together? i. As the time since the pebble hit the water increases, how does the radius of the circle change? As the amount of time since the pebble hit the water increases, the radius of the circular ripple increases. ii. As the radius of the circle increases, how does the area of the circle change? As the radius of the circular ripple increases, the area enclosed by the ripple increases. iii. As the time since the pebble hit the water increases, how does the area change? As the amount of time since the pebble hit the water increases, the area enclosed by the ripple increases

6 c. What quantities did the original question ask you to relate? What quantities in the situation are known? The original question asks that the time since the pebble hit the water be related to the area of the circle formed from the ripple traveling outward. Known quantities: the rate at which the radius is growing: 0.5m/sec. Unknown quantities: time since the pebble hit the water, radius of the ripple, area of the ripple. 114 e. Define the following functions. i. f that defines radius r (measured in meters) as a function of time t (measured in seconds). f (t) = 0.5t where r = f (t) ii. g that defines area A (measured in square meters) as a function of radius r (in meters). g(r) = πr 2 where A = g(r) f. Graph f and g and use these graphs to determine the area 3 seconds after the pebble hits the water. Explain your process

7 Radius of the circular ripple f Seconds since the pebble hit the water. 116 Area enclosed by the ripple (square meters) g Radius of the circular ripple 117 7

8 Radius of the circular ripple f Seconds since the pebble hit the water. 118 Area enclosed by the ripple (square meters) 3 seconds since the pebble hit the water the area enclosed by the circular ripple is about 7.07 square meters. g Radius of the circular ripple 119 8