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FUNCTION COMPOSITION: CHAINING TOGETHER
TWO FUNCTION PROCESSES
1. Noelle likes to plan out her meals ahead of time so that
she can purchase everything she needs at the grocery
store all at once. Because she likes Asian food so much,
one of her big expenditures is rice, which she purchases
from an Asian market by the pound. Noelle knows that
she can get 2.25 servings of rice for every 1 pound of
rice she purchases.
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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)
2
5
8.5
12.25
n
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1. Noelle likes to plan out her meals ahead of time so that
she can purchase everything she needs at the grocery
store all at once. Because she likes Asian food so much,
one of her big expenditures is rice, which she purchases
from an Asian market by the pound. Noelle knows that
she can get 2.25 servings of rice for every 1 pound of
rice she purchases.
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, $4.34
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1. Noelle likes to plan out her meals ahead of time so that
she can purchase everything she needs at the grocery
store all at once. Because she likes Asian food so much,
one of her big expenditures is rice, which she purchases
from an Asian market by the pound. Noelle knows that
she can get 2.25 servings of rice for every 1 pound of
rice she purchases.
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.
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1. Noelle likes to plan out her meals ahead of time so that
she can purchase everything she needs at the grocery
store all at once. Because she likes Asian food so much,
one of her big expenditures is rice, which she purchases
from an Asian market by the pound. Noelle knows that
she can get 2.25 servings of rice for every 1 pound of
rice she purchases.
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 $2.18.
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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)
2
5
8.5
12.25
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1. Noelle likes to plan out her meals ahead of time so that
she can purchase everything she needs at the grocery
store all at once. Because she likes Asian food so much,
one of her big expenditures is rice, which she purchases
from an Asian market by the pound. Noelle knows that
she can get 2.25 servings of rice for every 1 pound of
rice she purchases.
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):
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2. 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.
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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.
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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.
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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.
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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) = πr2 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.
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Radius of
the
circular
ripple
f
Seconds since the pebble hit the water.
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g
Area
enclosed
by the
ripple
(square
meters)
Radius of the circular ripple
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Radius of
the
circular
ripple
f
1.5
3
Seconds since the pebble hit the water.
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Area
enclosed
by the
ripple
(square
meters)
g
3 seconds since the pebble hit
the water the area enclosed by
the circular ripple is about
7.07 square meters.
7.07
1.5
Radius of the circular ripple
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