Precalculus  Exam  Questions:  Modules  2-­‐3    

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Test Instructions 1. Please read each problem carefully.

2. Fill out Scantron with your information. Be sure to include your ID Number.

3. Mark all answers clearly on the scantron sheet provided.

4. For open-ended questions, answers without supporting work will be given zero

credit. Partial credit is granted only if work is shown. 5. No calculators with Qwerty keyboards or ones like the Casio FX-2, TI-89 or TI-92

that do symbolic algebra may be used. 6. Proctors reserve the right to check calculators.

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allow your cell phone to ring while you are taking the exam. Do not use the calculator on your cell phone. If a proctor sees you using a cell phone, they will take your exam away from you.

 

Multiple  Choice  Section  (Worth  3  points  each)  –  Mark  all  answers  on  this  exam.  Show  work  on  this  exam  to  receive  partial  credit  for  incorrect  answers.    

 1. A  15-­‐inch  candle  burns  1.2  inches  per  hour.  If  h  =  the  number  of  hours  that  the  

candle  has  been  burning  and  L  =  1.2  h,  what  does  the  variable  L  represent  in  this  formula?      a. L  =  the  rate  at  which  the  candle  is  burning  after  h  hours  b. L  =  the  final  length  of  the  burning  candle  in  inches  c. L  =  the  length  of  the  candle  in  inches  that  remains  after  h  hours  d. L  =  the  number  of  inches  that  has  burned  from  the  candle  after  h  hours  e. L  =  the  length  of  the  part  of  the  candle  that  has  burned  (in  inches)  when  it  

stops  burning    2. Millie  ran  at  a  constant  speed  of  3  miles  per  hour  as  she  ran  4  laps  on  a  track.    

Which  of  the  following  describes  a  varying  quantity  in  this  situation?    a. The  distance  of  one  lap    b. The  elapsed  time  since  Millie  began  running  c. The  number  of  laps  Millie  had  run  when  she  was  finished  running  d. The  speed  that  Millie  was  running  e. The  time  it  took  Millie  to  run  one  lap.      

 3.   Suppose  r  changes  at  a  constant  rate  of  2.5  with  respect  to  p.  What  does  this  mean  for  

any  change  in  p?    

  a.        The  change  in  p  is  increasing  at  a  constant  rate  of  2.5     b.     The  change  in  r  is  always  2.5  times  more  than  the  change  in  p     c.   The  change  in  p  is  always  2.5  times  more  than  the  change  in  r  

d.       The  change  in  r  is  always  2.5  times  as  much  as  the  change  in  p  e.     The  change  in  p  is  always  2.5  times  as  much  as  the  change  in  r  

 4.   Jim  rides  his  bike  at  a  constant  rate  of  change  21.5  miles  per  hour  as  he  travels  from  his  

home  to  a  coffee  shop.  Read  all  five  options  and  select  the  one  that  includes  all  correct  answers.      

 a. The number of miles Jim travels is always 21.5 times as large as the number of hours Jim

rides. b. If Jim rides ½ of one hour he will ride ½ of 21.5 miles. c. For every 1 hour of elapsed time Jim travels a distance of 21.5 miles. d. All of the above e. A and C only

5. After  completing  a  pit  stop,  a  racecar  pulls  back  onto  the  racetrack.  Let  f  (t)  be  the  distance  of  the  racecar  t  seconds  since  leaving  the  pit.  What  is  the  meaning  of  f(5)  –  f(3)?  a. The  distance  traveled  by  the  car  from  t  =  3  to  t  =  5  seconds  since  leaving  the  

pit  b. The  average  speed  of  the  car  from  t  =  3  to  t  =  5  seconds  since  leaving  the  pit  c. The  increase  in  speed  as  the  car  moved  from  the  3  m  mark  to  the  5  m  mark  d. The  average  speed  of  the  car  as  the  car  moved  from  the  3  m  mark  to  the  5  m  

mark  e. The  time  elapsed  as  the  car  moved  from  the  3  m  mark  to  the  5  m  mark  

   (Use  the  following  statement  to  answer  questions  6  &  7)  Cameron  cycles  along  a  stretch  of  a  cycling  path  at  a  constant  rate  of  23  ft/sec.    6.    How  many  feet  does  Cameron  travel  in  4.3  sec?  

a. 5.35  ft  b. 18.7  ft  c. 27.3  ft  d. 98.9  ft  e. None  of  these  

 7.    Which  formula  determines  Cameron’s  distance  d  from  a  milepost  (in  feet),  given  the  amount  of  time  t  (in  seconds)  since  Cameron  passed  the  milepost?    

a. d=  t  /  23  b. t  =  d  +  23  c. d  =  23t  d. t  =  23d  e. d  =  t  +  23  

 8.   A  spherical  snowball  originally  has  a  radius  of  8  cm.  As  it  melts  the  radius  decreases  at  a  rate  of  0.6  cm/minute.  Which  of  the  following  defines  the  volume  of  the  snowball  (in  cm3)  as  a  function  of  the  amount  of  time  (in  minutes)  since  the  snowball  began  melting?  (Recall  that  the  volume  of  a  sphere  is  determined  by  

V =43πr3 ).  

a. g(t) = 8 − 0.6t  

b. g(t) = 43πr3t  

c. g(t) = 43π 8 − 0.6t 3( )  

d. g(t) = 43π 8 − 0.6t( )3  

e. g(r) = 43πr3  

 9.  The  distance,  s  (in  feet),  traveled  by  a  car  moving  in  a  straight  line  is  given  by  the  function,   ( ) ttts += 2 ,  where  t  is  measured  in  seconds.    Find  the  average  velocity  for  the  time  period  from  t  =  1  to  t  =  4.  

a. sec5 ft  b. sec6 ft  c. sec9 ft  d. sec10 ft  e. sec11 ft  

 10. The graph below represents the number of points scored by a basketball team, P, as a

function of the number of minutes after the start of the game. Evaluate P(15) and explain its meaning.

a. P(15) = 10; Fifteen minutes after the start of the game the team had scored 10 points. b. P(15) = 10; Ten minutes after the start of the game the team had scored 15 points. c. P(15) = 20; Fifteen minutes after the start of the game the team had scored 20 points. d. P(15) = 20; Twenty minutes after the start of the game the team had scored 15 points. e. P(15) = 8; Fifteen minutes after the start of the game the team had scored 8 points.

                                     11.  A  word  processor  determines  the  width  of  the  body  of  text  on  a  page  for  a  margin  

setting  of  x  inches.  If  the  page  is  originally  8.5  inches  wide  and  has  two  equal  size  margins  on  each  side,  which  formula  determines  the  width  of  the  body  of  the  text?    

 a.    b.    c.    d.    e.    2x  

2x − 8.5x − 8.58.5 − 2x8.5 − x

   

12.    Use  the  graph  of  f  to  solve  f  (x)  =  −3  for  x.      

a.    (−3,  −2)  b.   −4  c.   (−4,  −3)  d.   −2  e.   −3  

  13.  The  variables  x  and  y  in  the  table  below  are  in  

a  linear  relationship.       What  is  the  value  of  y  when  x  is  9?       a.   0     b.   7     c.   1     d.   –1     e.   –4      14.  Suppose  you  determine  the  average  speed  of  this  racecar  over  the  interval  t  =  1  

to  t  =  4  sec.  Which  statement  best  describes  the  meaning  of  average  speed  in  this  context?  

 a. The  sum  of  the  initial  and  final  speeds  divided  by  2      b. The  speed  that  the  racecar  drove  most  of  the  time.  c. The  constant  speed  that  the  racecar  traveled  during  the  3  seconds.  d. The  constant  speed  needed  to  travel  the  same  distance  as  the  racecar  in  the  

same  amount  of  time  (3  seconds).  e. The  sum  of  the  distances  at  t  =  1,2,3,4  divided  by  4  

   15. If  S(m)  represents  the  salary  (per  month),  in  hundreds  of  dollars,  of  an  

employee  after  m  months  on  the  job,  what  would  the  function  R(m)  =  S(m  +  12)  represent?  

 a. The salary of an employee after 12 months on the job. b. $12 more than the salary of someone who has worked for m months. c. An employee who has worked for m + 12 months. d. The salary of an employee after m + 12 months on the job. e. Not enough information.

       

x   y  1   15  3   11  9    12   –7  

Use  the  graphs  of  f  and  g  to  answer  items  16  and  17.   16.  Use  the  graphs  of  f  and  g  to  evaluate   .       a.   –2     b.   1     c.   3     d.   4     e.   Not  defined      17. Evaluate . a. -4   b.   -­‐2     c.   0     d.   2     e.   4    18.  Let  S(m)  represents  the  salary  (per  month),  in  hundreds  of  dollars,  of  an  employee  after  

m  months  on  the  job.  What  does  the  solution  to  the  equation  S(m)  =  23  represent?       a.   The  salary  of  an  employee  after  23  months  on  the  job.     b.   The  monthly  salary  of  the  employee  is  23.     c.   The  number  of  months  it  would  take  for  an  employee  to  earn  a  salary  of  $2,300  per  

month.     d.   The  monthly  salary  of  the  employee  is  $2,300.     e.     The  number  of  months  it  would  take  for  an  employee  to  earn  a  salary  of  $23  per  

hour.      19. Given  the  function    and   ,  evaluate   .    

a. 10  b. 11  c. 20  d. 25  e. 36  

 20.  Given  that  f(1)  =  6  and  f(5)  =  14,  what  is  the  average  rate  of  change  of  f  on  the  interval       from  x  =  1  to      x  =  5.       a.   10     c.   8     c.   2     d.   4     e.   6    

   

g f 2( )( )

f 2( ) − g 0( )

h x( ) = 3x −1 g x( ) = x2 g h 2( )( )

21. What  is  the  domain  of  the  following  function:   ( ) ?12

−+=

xxxf

         

a. x >1  b. 1≠x  c. x > −2 and x ≠ 1  d. x > −2    e. All real numbers

22.  At  5:00  am  Maria  leaves  a  gas  station  and  drives  on  I-­‐70  toward  Denver  at  a  constant  rate  of  40  miles  per  hour.  At  5:30  am  Robert  leaves  the  same  gas  station  and  also  drives  on  I-­‐70  towards  Denver  at  a  constant  rate  of  60  miles  per  hour.  At  6:00  am  how  far  apart  are  Maria  and  Robert?    

  a.   0  miles     b.   10  miles     c.   20  miles     d.   30  miles     e.   40  miles  

23. A  baseball  card  increases  in  value  according  to  the  function,  b(t)  =  52  t  +  100,  where  b  

gives  the  value  of  the  card  in  dollars  and  t  is  the  time  (in  years)  since  the  card  was  purchased.    Which  of  the  following  describe  what   2

5  conveys  about  the  situation?    

I. The  card’s  value  increases  by  $5  every  two  years.                      II.      Every  year  the  card’s  value  is  2.5  times  greater  than  the  previous  year.  

II. The  card’s  value  increases  by   25 dollars  every  year.  a. I  only  b. II  only  c. III  only  d. I  and  III  only  e. I,  II  and  III  

   24. Two  cars  that  are  originally  216  miles  apart  start  traveling  toward  one  another  at  9  am.  

One  car  travels  62  miles  per  hour  and  the  other  travels  68  miles  per  hour,  both  with  their  cruise  control  on.  Which  formula  represents  d,  the  distance  between  the  two  cars  t  hours  after  9  am.    

a. d  =62 68t t+    

b. d  = 216− 62t          

c. d  =68 62t t−              

d. d  = 216 (68 62 )t t− +    

e. d  = 216+ (68t + 62t)    

25. Given  that   5 10Q P= + ,  which  of  the  following  is  correct?    

a.              

b.                  

c.                

d.  

A racecar travels around a circular track for 2083 seconds. Let d represent all possible values of the distance of the racecar (in meters) from the starting line, and let t represent all possible values of the number of seconds since the racecar left the starting line. The relationship between d and t is given by the formula f(t) = 0.6t2 + 2t where d = f(t). 26. What is the meaning of f(5) – f(3)?

f. The time elapsed as the car moved from the 3 m mark to the 5 m mark g. The distance traveled by the car from t = 3 to t = 5 seconds h. The increase in speed as the car moved from the 3 m mark to the 5 m mark i. The average speed of the car from t = 3 to t = 5 seconds e. None of the above

27. Evaluate f(5) – f(3).

a. 6.4 b. 11.4 c. 13.6 d. 25 e. None of the above

28. What is the average speed of the racecar as the number of seconds since it left the starting

line increases from 1 to 4 seconds?

a. 3 m/s b. 5 m/s c. 15 m/s d. 17.6 m/s e. None of the above

29. Which statement best describes the meaning of average speed in this context?

f. The sum of the distances at t = 1, 2, 3, 4 divided by 4 g. The speed that the racecar drove most of the time. h. The constant speed that the racecar driver traveled during the 3 seconds. i. The constant speed needed to travel the same distance as the given car in the same

amount of time (3 seconds). j. The sum of the initial and final speeds divided by 2

See  open-­‐ended  questions  on  next  page

P = 15Q +10

P = 15Q −10

P = 15Q − 2

P = 15Q +10

Open-­‐Ended  Section  –  Show  all  work  and  explain  all  answers  thoroughly  on  this  exam.  Be  sure  to  define  variables  where  needed.    

30.  (12  pts)    A  large  spool  is  used  to  hold  rope  that  is  wound  around  the  spool.    The  more  rope  wound  around  the  spool,  the  greater  the  combined  weight  of  the  spool  and  rope.    The  graph  below  shows  that  when  5  feet  of  rope  is  wound  around  the  spool,  the  total  weight  of  the  spool  and  rope  is  3.95  pounds.    Note  that  the  rope  weighs  0.27  pounds  per  foot.    

   

   

a. Suppose  the  number  of  feet  of  rope  on  the  spool  increases  from  the  given  point  to  8.4  feet.    What  is  the  change  in  the  number  of  feet  of  rope  on  the  spool?              

b. Represent  this  change  on  the  graph  above.  Explain  the  thinking  you  used  to  represent  this  change.            

c. What  is  the  total  weight  of  the  spool  and  rope  when  there  are  8.4  feet  of  rope  on  the  spool?    Explain  how  you  determined  your  answer.                  

d. What  is  the  weight  of  the  spool  without  any  rope?    Explain  how  you  determined  this  value  and  represent  your  reasoning  on  the  graph  above.  

Total length of rope on the spool (in feet)2 4 6 8 12

Tota

l wei

ght o

f spo

ol a

nd ro

pe (i

n po

unds

)

1

2

3

4

5

(5, 3.95)

31.   A  3  m  x  1.5  m  piece  of  plywood  is  being  used  to  build  an  open-­‐top  toy  chest.  The  chest  is  formed  by  making  equal-­‐sized  square  cutouts  from  two  corners  of  the  plywood  at  the  ends  of  a  3-­‐meter  side.  After  these  squares  are  discarded,  three  more  cuts  are  made  (at  the  dashed  lines  on  the  figure)  and  these  pieces  are  “folded  up”  and  secured  to  create  three  of  the  four  vertical  sides.    When  the  open  side  is  placed  against  a  wall,  the  open  top  toy  chest  is  formed.    Neglect  the  thickness  of  the  wood  when  answering  these  questions.  

   

a.     If  the  variable  x  represents  the  length  of  the  sides  of  the  square  cutouts  in  meters,  define  a  function  f    that  expresses  the  volume  of  the  toy  chest  (in  cubic  meters)  as  a  function  of  the  length  of  the  side  of  the  square  cutouts,  x.      

         

b.     What  are  the  possible  values  that  x  can  assume  in  the  context  of  this  problem?      

       c.     As  the  length  of  the  side  of  the  square  cutout  increases  from  0.5  meters  to  

0.75  meters,  how  does  the  volume  of  the  toy  chest  change?  (You  do  not  need  to  calculate  the  answer.  You  may  just  write  the  expression  that  you  would  use  to  calculate  the  answer.)  

         

32. The air temperature, T, in degrees Fahrenheit, is given in terms of the chirp rate, R, in chirps per minute, of a snowy tree cricket by the function f where 𝑇 = 𝑓 𝑅  and  𝑓(𝑅) = !

!𝑅 + 40.

a. Determine the function rule for the inverse function 𝑓!!

b. What does the inverse function represent in this situation?  

Additional  Skill  Problems:    

33. If f x( ) = 6x2 −1

x − 2 find f 0( ) .

34. If g x( ) = 4x − 5 , find g 2 + h( ) − g 2( )

h. Simplify your answer as far as possible.

35. If h x( ) = x2 − 3x and k x( ) = x +1 , find h k 8( )( ) .

36. If y = l x( ) = 4x + 72x

, find l−1 y( ) .

37. If R x( ) = 300x and C x( ) = 200 +175x , find the value of x where R x( ) = C x( ) .