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"'~ Functions Applications Worksheet

Name

1.) Hippopotamus Problem: In order to hunt hippopotami, a hunter must have a hippopotamushunting license. Since the hunter can sell the hippos he catches, he can use the proceeds topay 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 WildlifeCommission allows a limit of 10 hippos per hunter. Let h be the number of hippos caught, andlet d be the number of dollars profit made. Assume that hand d are related by a linearfunction. (3 -L6S1JJ(« d) /

I (7, Iss-o)a) Which variable should be dependent and whic::Yhould be independent? •

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b) Write a suitable domain for the independent variable. {;OJ ,! 2.} ••• I /0)

c) Write the particular equation expressing the dependent variable in terms of theindependent variable. d( "\_ .

, . _':f(d - 90() IL - ~7 SO

d) Plot the graph of this function, observing the domain you wrote in part b.

d~~~ (0/ -¥7~-oJd(o) ~9oo(o) - L/7S-0

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0= 1ot) -It.. -47SO

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~ d.(~) lhcZU.-rtd,Ck) r::. 9()o (0) - q7S-o

( IJ I -L/7S-0)

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e) Calculate the dJand h- intercepts. Tell what each means in the real world.d.-~ (0,-'17$7») ~ ~.;::

2.) Cricket Problem: Based on information in Deep River Jim's Wilderness Trailbook, the rate atwhich 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. (~J cJ •.:~.;) (.r~ 7(.,)

(6S"; 100)a) Write the particular equation expressing chirping rate in terms of temperature.

C. ( t);: ~S" t - 'Ib) Predict the chirping rate for gOoF. C C9tJ) = J7s(90)-¥ C( fa):: I!/o ~

c) How warm is it if you count 120 chirps per minute? /~O = %-t- t/- t ::77.S-°F'

d) Calculate the temperature-intercept. What does this number tell you about the realworld? t) = Ills- t -If t" = a. s°!=

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?~+71 ~~ etA! ~ rvJ.'1 A~ _

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3.) Ice Cube Problem: You run some water into a pitcher, then cool it down by adding icecubes. You find that putting in 5 cubes cools the water down to 76°F. Putting in 10 more cubescools down the water to 48°F. Assume that the temperature of the water varies linearly with thenumber of cues you put in.

a) Write the particular equation expressing temperature in terms of number of cubes.Show work. (~I ~ •....•.z; .•..v) (S; 7') (IS/ L/a)

I?? = '''-48'<: ~f'.s-- / S"" - /0

b) Predict the temperature to which the water would cool if you had put in a total of 7cubes. Show work.

c) What total number of cubes would have to be put in to reduce the water temperature tofreezing (32°F)?

.3-l c: -1"1s- c of- 90 c. -:::~ 0, 71'1 ~ .:? I C.4.~

d) What is the temperature-intercept? What does this number represent in the real world?t (c) -= -1'-1/;.( o) +- '1tJ t (C) =Yo" ~ U:=t-=... l.•.. ...) 1. ~

Domain: .__ -L- _ Range:,_(=--3_l_d

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4.) Terminal Velocity Problem: If you jump out of an airplane at high altitude, but do not open yourparachute, you will soon be falling at a constant velocity called your "terminal velocity." Suppose thatat time t = 0 you jump. When t = 15 seconds, your wrist altimeter shows that your distance from theground, d, is 3600 meters. When t =35, you have dropped to d = 2400 meters. Assume that youhave already reached your terminal velocity by the time t = 15. ( ~ I h r~ )

a) Explain why d varies linearly with t after you have reached your terminal velocity.

(IS; 3'00) . - ~ _b) Write the parttcular equanon expressmq d in terms of t. d {t- J :::: -~() t -I- 4J "()tJ

(jS",2.flof» 0 -=-'ot+LlS""Ooc) If you neglect to open your parachute, when will you hit the ground? ~ 76'~

m,z -'0d) According to your linear model, how high was the airplane when you jumped?

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e) The airplane was actually at 4200 meters when you jumped. How do you reconcile thisfact with your answer to part d?

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f) Sketch a reasonable gra~h LJd versus t, showing the linear part, the part before youreached terminal velocity, and the part after you open your parachute .

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1000

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3000

10 20 30 (jO Sl> "0 70 8o,j~

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g) What was your terminal velocity in meters per second? In kilometers per hour?Show work.

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5.) Thermal Expansion Problem: Bridges on expressways often have expansion joints, which aresmall gaps in the roadway between one bridge section and the next. The gaps are put there so thatthe bridge will have room to expand when the weather gets hot. Suppose that a bridge has a gap of1.3 cm when the temperature is 22°C, and that the gap narrows to 0.9 cm when the temperaturewarms to 30°C. Assume that the gap width varies linearly with the temperature. (~) d~)

a) Write the particular equation for gap width as a funciton of temperature. Co?.:l/ I. 3)

1(t):= - fa -t +- % ( .3°1 ,,9)hi:: 1.3-,9_.4" __ (

b) How wide would the gap be at 35°C? At -1DOC? ~~ - 30 - =i'" i:Oj{3s-) e: ., S'" e- J{ -10) = ~. C; c...-

c) At what temperature would the gap close completely? What mathematical name isgiven 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. Use an appropriate domain.

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6.) Celsius-to-Fahrenheit Temperature Conversion: The Fahrenheit temperature, F, and theCelsius 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. ( C J J:J (0) 32J

a) Write an equation expressing F in terms of C. _F__C"---=:s:.....:_C_+_3_.:J..._· _

b) Transform the equation so that C is in terms of F. __ C_c_,-=-f_{_F_-_3_d_} _

c) Lead boils at 1620oC. What Fahrenheit temperature is this? Which form of the equationis more appropriate to use in answering this question?

F s: ~(I'~O) -i-.3~ ~fj'tfR °c

31°cd) Normal body temperature is 98.6°F. What Celsius temperature is this? _

e) If the weather forecaster says it will be 40°C today, will it be hot, cold, or medium?Explain.

~&: ~(4oJ+-32. -=z/oL/I:>F ~

f) The coldest possible temperature is absolute zero, -273°C, where molecules stopmoving. What Fahrenheit temperature is this?

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g) For what temperature is the number of Fahrenheit degrees equal to the number ofCelsius degrees?

h) Sketch the graph of F as a function of C, showing clearly the domain implied by part fand 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 herfoot. The smallest adult shoe is size 5, and fits a 9-inch long foot. An 11-inch foot takes a size 11shoe.

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? __ 1-'1'--- _

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 impliedat 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 thehighway. As you drive, the number of minutes, t, since you had the tank filled, and the number ofliters, 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. I).,·, ~ .."f".4..c.c-

d) Find the g-intercept and tell what it represents in the real world.

7& I ~ .I~ w-..-I 7(; I c: 'I d~ ~ ~ ~e) Sketch the graph of this linear function. ~ ~ lZ/o =r=:

If) Tell what the slope represents in the real world and tell the significance of the factthat the slope is negative.

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