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Functions Applications Worksheet
1.) Hippopotamus Problem: In order to hunt hippopotami, a hunter must have a hippopotamus
hunting license. Since the hunter can sell the hippos he catches, he can use the proceeds to
pay for part or all of the cost of the license. If he catches only 3 hippos, he is still in 'debt by
$2050. If he catches 7 hippos, he makes a profit of $1550. The African Game and Wildlife
Commission allows a limit of 10 hippos per hunter. Let h be the number of hippos caught, and
let d be the number of dollars profit made. Assume that hand d are related by a linear
function.
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a) Which variable should be dependent and whic::Yhould
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be independent?
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b) Write a suitable domain for the independent variable.
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c) Write the particular equation expressing the dependent variable in terms of the
independent variable. d( " \ _
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d) Plot the graph of this function, observing the domain you wrote in part b.
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e) Calculate the dJand h- intercepts. Tell what each means in the real world.
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2.) Cricket Problem: Based on information in Deep River Jim's Wilderness Trailbook, the rate at
which crickets chirp is a linear function of temperature. At 50°F they make 76 chirps per minute,
and at 65°F they make 100 chirps per minute.
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a) Write the particular equation expressing chirping rate in terms of temperature.
C. ( t);:
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b)
Predict the chirping rate for gOoF. C
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c) How warm is it if you count 120 chirps per minute?
d)
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Calculate the temperature-intercept.
What does this number tell you about the real
world?
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e) Sketch the graph of this function in a reasonable domain.
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f) What does the chirping-rate intercept tell you about the realworld?
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3.) Ice Cube Problem: You run some water into a pitcher, then cool it down by adding ice
cubes. You find that putting in 5 cubes cools the water down to 76°F. Putting in 10 more cubes
cools down the water to 48°F. Assume that the temperature of the water varies linearly with the
number of cues you put in.
a) Write the particular equation expressing temperature in terms of number of cubes.
Show work.
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b) Predict the temperature to which the water would cool if you had put in a total of 7
cubes. Show work.
c) What total number of cubes would have to be put in to reduce the water temperature to
freezing (32°F)?
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d) What is the temperature-intercept? What does this number represent in the real world?
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4.) Terminal Velocity Problem: If you jump out of an airplane at high altitude, but do not open your
parachute, you will soon be falling at a constant velocity called your "terminal velocity." Suppose that
at time t = 0 you jump. When t = 15 seconds, your wrist altimeter shows that your distance from the
ground, d, is 3600 meters. When t =35, you have dropped to d = 2400 meters. Assume that you
have already reached your terminal velocity by the time t = 15.
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a) Explain why d varies linearly with
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t after
you have reached your terminal velocity.
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b) Write the parttcular equanon expressmq d in terms of t.
d {t- J :::: -~() t
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c) If you neglect to open your parachute, when will you hit the ground?
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d) According to your linear model, how high was the airplane when you jumped?
e) The airplane was actually at 4200 meters when you jumped.
fact with your answer to part d?
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How do you reconcile this
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f) Sketch a reasonable gra~h L d versus t, showing the linear part, the part before you
reached terminal velocity, and the part after you open your parachute .
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1000
10
20
30
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g) What was your terminal velocity in meters per second?
Show work.
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In kilometers per hour?
5.) Thermal Expansion Problem: Bridges on expressways often have expansion joints, which are
small gaps in the roadway between one bridge section and the next. The gaps are put there so that
the bridge will have room to expand when the weather gets hot. Suppose that a bridge has a gap of
1.3 cm when the temperature is 22°C, and that the gap narrows to 0.9 cm when the temperature
warms to 30°C. Assume that the gap width varies linearly with the temperature. (~)
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a) Write the particular equation for gap width as a funciton of temperature.
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c) At what temperature would the gap close completely?
given to this temperature?
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d) Would the temperature ever be likely to get hot enough to close the gap?
Justify your answer.
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e) Sketch the graph of this linear function.
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b) How wide would the gap be at 35°C? At -1DOC?
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Use an appropriate domain.
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6.) Celsius-to-Fahrenheit Temperature Conversion: The Fahrenheit temperature, F, and the
Celsius temperature, C, of an object are related by a linear function. Water boils at 100°C or 212°F,
and freezes at O°C or 32°F.
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a) Write an equation expressing F in terms of C.
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b) Transform the equation so that C is in terms of F. __
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C_c_,-=-f_{_F_-_3_d_}
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c) Lead boils at 1620oC. What Fahrenheit temperature is this? Which form of the equation
is more appropriate to use in answering this question?
F
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d) Normal body temperature is 98.6°F. What Celsius temperature is this?
31°c
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e) If the weather forecaster says it will be 40°C today, will it be hot, cold, or medium?
Explain.
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f) The coldest possible temperature is absolute zero, -273°C, where molecules stop
moving. What Fahrenheit temperature is this?
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g) For what temperature is the number of Fahrenheit degrees equal to the number of
Celsius degrees?
h) Sketch the graph of F as a function of C, showing clearly the domain implied by part f
and the F-intercept.
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7.) Shoe Size Problem: The size of shoe a person needs varies linearly with the length of his or her
foot. The smallest adult shoe is size 5, and fits a 9-inch long foot. An 11-inch foot takes a size 11
shoe.
a) Write the particular equation expressing shoe size in terms of foot length .
.f(L) ;- 3L -..11.
b) If your foot is a foot long, what size shoe do you need? __
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c) Bob Lanier, who once played basketball for the Detroit Pistons, wears a size 22 shoe.
How long is his foot?
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d) Plot the graph of adult shoe size versus foot length. Be sure to observe the domain implied
at the beginning of this problem.
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8.) Gas Tank Problem: Suppose that you et your car's gas tank filled up, then drive off down the
highway. As you drive, the number of minutes, t, since you had the tank filled, and the number of
liters, g, remaining in the gas tank are related by a linear function.
a) Which variable should be indepe~dent, and which should be dep?ndent?
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b) After 40 minutes you have 52 liters left. An hour after the fill-up you have 40 liters left.
Write the particular equation for this function.
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c) Use the equation to predict the time when you will run out of gas.
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d) Find the g-intercept and tell what it represents in the real world.
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e) Sketch the graph of this linear function.
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f) Tell what the slope represents in the real world and tell the significance of the fact
that the slope is negative.
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