# College Algebra Extra Credit Problems

For each problem, a correct solution with appropriate
support work is worth one point.

1. The growth of a human fetus more than 12 weeks old can be approximated by the
formula

L = 1.53t - 6.7, where L is the length (in centimeters) and t is the age (in
weeks). Prenatal length can be

determined by ultrasound. Approximate the age of a fetus whose total length is
28 centimeters. Show

all of your work or explain how you came up with your solution.

2. In 1990, around 115 billion dollars was invested globally in
telecommunications infrastructure. In 1994,

around 145 billion was spent. Assuming that global investment in telecom
infrastructure is growing at a

constant rate, write a linear model which gives investment (I) in billions of
dollars as a function of t, the

number of years since 1990. Use your model to predict global telecom investment
in 2003.

3. The following table gives the enrollment at NWACC for the Fall semesters from
1990 to 2000.

Create a scatter plot of the data. Assuming that enrollment is growing at a
constant rate, use the

regression capabilities of your calculator to run a linear regression analysis
on the data. Let x = 0

represent 1990. What is the linear equation which best fits the data (round to
three decimal places)?

What is the correlation coefficient? According to this model, what will
enrollment be in the Fall of

2005? How confident are you about this projection?

Year | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |

Enrollment | 1232 | 1632 | 1889 | 1972 | 2037 | 2244 | 2941 | 3240 | 3542 | 3923 | 4095 |

4. It is a surprising biological fact that most crickets
chirp at a rate that increases as the temperature

increases. For the snowy tree cricket (Oecanthus fultoni), the relationship
between temperature and

chirp rate is so reliable that this type of cricket is called the thermometer
cricket. One can estimate the

temperature (in °F) by counting the number of times a snowy tree cricket chirps
in 15 seconds and

adding 40. For instance, if we count 20 chirps in 15 seconds, then a good
estimate of the temperature

would be 20 + 40 = 60°F. The rule used to find the estimated temperature T (in
°F) based on the

measured chirp rate R (in chirps per minute) is an example of a function.
Describe this function in four

different ways: using words, a table, a graph, and a formula.

5. The difference quotient
can be used to evaluate the average velocity
of an object rolling

down a ramp. s is the distance the object travels, t is the time in seconds
after the object is released, a is

the start time, and Δt is the length of the time interval (finish time - start
time). If the distance traveled

by a ball rolling down a ramp is given by the function s (t) = 4t^{2},
what is the average velocity of the ball

over the time interval from t = 3 to t = 3.01? Show all of your work or explain
how you came up with

your solution.

6. The concentration C(t), in milligrams per liter, of a
medication in a patient’s blood is given by the

function where t is time in hours after the medication was taken. Use the C(t) =
25t^{3} - 150t^{2} + 225t

difference quotient to find the average rate
of change of the concentration over the

time interval from t = 1 to t = 1.25. After how many hours will the drug be
totally eliminated from the

bloodstream? Show all of your work or explain how you came up with your
solutions.

7. Captain Janeway left the holodeck at 7:45 to meet Tuvok, her Chief of
Security, on the Bridge. After

walking for 3 minutes, she realized she had forgotten her tricorder and returned
to get it. She picked up

the tricorder and resumed her walk, arriving on the Bridge at 8:00. After 15
minutes the discussion was

over, and Janeway returned to the holodeck. Graph Janeway’s distance from the
holodeck as a function

of time. Determine on which time intervals the graph is increasing, decreasing,
and constant.

8. The velocity of the air that is expelled during a cough can be modeled by v
=.6r^{2} - r^{3} , where v is

measured in centimeters per second and r is the radius of the trachea in
centimeters. Find the radius of

the trachea, to the nearest .1 centimeter, that maximizes the velocity of the
air. Physical considerations

of the structure of the trachea indicate that .3 ≤ r ≤ .6 is an appropriate
domain for r. Show all of your

work or explain how you came up with your solution.

9. A coed soccer team has 16 players. Write a function g that describes the
number of girls on the team if

x of the players are boys. If f(x) = x, what transformations of function f would
be necessary to obtain

function g? Show all of your work or explain how you came up with your solution.

10. A fire has started in a dry open field and is spreading in the form of a
circle. If the radius of the circle

increases at the rate of 6 ft/min, express the total fire area A as a function
of time t (in minutes). Show

all of your work or explain how you came up with your solution.

11. In harvesting 10% of a rainforest, logging companies will often destroy 50%
of the unharvested area just

to get to the valuable trees. If the function u = .9x gives the unharvested area
with the valuable timber

removed, and function w = .5u gives the amount of unharvested area that is
destroyed and wasted, write

a function which gives the wasted area (w) as a function of the original area
(x).

12. The formula P = 20 + .4t gives the population (P, in thousands) of a certain
town as a function of time

(t = number of years since 1970 since t = 0 corresponds to 1970). Find the
inverse function. What does

it tell us?

13. In 1974, at the tomb of an emperor near Xian, Chinese workers found three
pits that contained life-size

sculptures of warriors. The largest rectangular pit has a length that is 40
yards longer than three times

its width. If the perimeter is 640 yards, what are the length and width of the
pit? Show all of your work

or explain how you came up with your solution.

14. A group of hikers from Tulsa hiked down into the Grand Canyon in 3 hours 30
minutes. Coming back

up on a trail that was 4 miles shorter, they hiked 2 mph slower and it took them
1 hour longer. What

was their rate going down? Show all of your work or explain how you came up with
your solution.

15. The cost of installing insulation in a particular two-bedroom house is
$1080. Present monthly heating

bills average $60, but the insulation is expected to reduce the heating cost by
10%. How many months

will it take to recover the cost of the insulation? Show all of your work or
explain how you came up

with your solution.

16. The equation gives
the total time (T) that elapses from the moment an object is dropped

into a well to the moment when the sound of its landing is heard. d is the depth
of the well in feet.

Use a graphing calculator to determine the depth when T = 4 seconds. Show all of
your work or

explain how you came up with your solution.

17. How is dividing by a complex number similar to the process of rationalizing
a denominator?

18. Michael Chang, a professional tennis player, skillfully uses the lob as both
a defensive and an offensive

shot. Chang knows that a well-placed lob can buy the time needed to prepare for
the next shot. How

long does it take for a tennis ball to hit the earth if it is hit straight
upward with a velocity of 60 feet per

second from a height of 5 feet? Show all of your work or explain how you came up
with your solution.

19. The speed of sound in air at 0°C (or 273 K) is 1087 ft/sec, but this speed
increases as the temperature

rises. The speed v of sound at temperature T in Kelvin is given by
At what

temperatures does the speed of sound exceed 1100 ft/sec? Show all of your work
or explain how you

came up with your solution.

20. The Guinness Book of World Records reports that German shepherds can make
vertical leaps over

10 feet when scaling walls. If the distance d (in feet) off the ground after t
seconds is given by the

equation , for how many seconds is the dog more than 9 feet off the ground? d =
-16t^{2} + 24t + 1

Show all of your work or explain how you came up with your solution.

21. A machinist is producing a circular cylinder on a
lathe. The circumference of the cylinder must be 28

inches, with a tolerance of .15 inch. What maximum and minimum radii (to the
nearest .001 inch) must

the machinist stay between to produce an acceptable cylinder? Show all of your
work or explain how

you came up with your solution.

22. The national catch of Atlantic Cod has been declining in the past few
decades. We can model this

decline with the equation where x is the number of years since 1950 and y =
-.003x^{2} + .120x + 1.909

is the catch in millions of metric tons. In which year did the catch reach its
maximum? In which year

will the catch reach 0? For what years was the catch increasing? Decreasing?
When was the catch

greater than 2.5 million metric tons? Show all of your work or explain how you
came up with your

solutions.

23. To determine the appropriate landing speed of an airplane, the formula is
used, where D = .1x^{2} - 3x + 22

x is the initial landing speed in feet per second and D is the distance needed
in feet. If the landing speed

is too fast, the pilot may run out of runway; if the speed is too slow, the
plane may stall. What is the

appropriate landing speed if the runway is 800 feet long? Show all of your work
or explain how you

came up with your solution.

24. The following table gives the number of motor vehicle thefts (in thousands)
in the U.S. for the years

1983 - 1993. x = 1 represents 1983. Use the regression capabilities of your
calculator to fit a cubic

model to this data. Round to three decimal places. Use your model to predict the
number of motor

vehicle thefts in 1995.

Year (x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

Thefts (y) | 1008 | 1032 | 1103 | 1224 | 1289 | 1433 | 1565 | 1636 | 1662 | 1611 | 1561 |

25. A security panel with n buttons is disarmed when four
different buttons are pushed in the correct order.

The total number of four-button sequences in which no button is pushed more than
one time is given by

P(n) = n^{4} - 6n^{3} +11n^{2} - 6n where n is an
integer greater than or equal to 4. For security reasons, the

manufacturer of the panels requires that P(n) ≥ 14,000. Find the least number of
buttons that will meet

the manufacturer’s requirement. Show all of your work or explain how you came up
with your

solution.

26. The town of Smallsville was founded in 1900. Its population y (in hundreds)
is given by the function

y = -.1^{4} + 1.7x^{3} -9x^{2} + 14.4x +5 where x is the
number of years since the foundation of Smallsville.

What was the population of Smallsville when it was founded? When did Smallsville
become a “ghost

town” with noone living in it? What was the largest population of Smallsville
after 1905? When did

Smallsville reach that population? Show all of your work or explain how you came
up with your

solutions.

27. Pollution in the Tangipahoa River has been blamed primarily on the area
dairy farmers. The formula

is used to model the cost in dollars to area
dairy farmers for removing p% of the coliform

bacteria from the river. If $1.2 million is spent, what percentage of the
coliform bacteria has been

removed? Show all of your work or explain how you came up with your solution.

28. A formula to determine the monthly payment (PMT) for a car loan, home
mortgage, or other installment

loan is given by where P (called the present
value) is the amount borrowed,

i is the annual interest rate, and n is the total number of payments. John
purchases a car and secures a

loan for $9000 at an annual interest rate of 10% for a term of 4 years. What is
his monthly car payment?

If he makes all 48 payments, how much money will he have paid? How much of this
total is interest?

Show all of your work or explain how you came up with your solution.

29. Sam Donahue has $8765 in his savings account on
January 1. The account pays 5.25% interest

compounded daily. On February 12, he deposited $936. Then on March 20, he
deposited a tax refund

check for $650. Find the balance at the end of the quarter and the interest
earned during the quarter.

Show all of your work or explain how you came up with your solution.

30. Astronomers use the distance modulus of a star as a method of determining
the star’s distance from

Earth. The formula is M = 5 log r - 5, where M is the distance modulus and r is
the star’s distance from

Earth in parsecs (One parsec is approximately 3.3 light-years or 2.1×10^{13}
miles.) How many parsecs

from Earth is a star with a distance modulus of 3? Show all of your work or
explain how you came up

with your solution.

31. Usually short-term interest rates (less than 2 years) are lower than
long-term rates (more than 10 years)

because short term securities are less risky than long-term ones. However, in
periods of high inflation,

the situation is reversed. During the early 1980s inflation was very high in the
U.S. The rates for short-

and long-term U.S. Treasury securities during 1980 are given in the following
table. Use this data and

the regression capabilities of your calculator to generate a logarithmic model
for the data. Then use

your model to predict the interest rate in 1980 for a 15 year T-bill. Show all
of your work or explain

how you came up with your solution.

Term (years) | 5 | 1 | 5 | 10 | 20 |

Rate (%) | 15 | 14 | 13.5 | 12.8 | 12.5 |

32. Industrial psychologist study employee training
programs to assess the effectiveness of the instruction.

In one study, the percent score P on a test for a person who has completed t
hours of training was given

by How many hours of training are
necessary to achieve a 70% on the test?

Show all of your work or explain how you came up with your solution.

33. Sodium-24 is a radioactive isotope of sodium that is used to study
circulatory dysfunction.

4 micrograms of sodium-24 is injected into a patient and 5 hours later, 3.178
micrograms remain.

Assuming that the isotope is dissipating exponentially, write an exponential
decay model which gives

the amount of sodium-24 (A) as a function of time (t) in hours. Use your model
to determine the amount

of isotope left after 10 hours. Show all of your work or explain how you came up
with your solution.

34. Congratulations, you’ve just won $2500! You decide to invest in a mutual
fund and save the money for

an emergency. After 3 years, you have $2932.12. What was the interest rate on
the fund if the interest

was compounded quarterly? Show all of your work or explain how you came up with
your solution.

35. Assuming that air resistance is proportional to the velocity of a falling
object, the velocity (in feet per

second) of an object t seconds after it has been dropped is given by
When will the

velocity be 70 feet per second? Show all of your work or explain how you came up
with your solution.

36. A detective discovered a body in a vacant lot at 7 a.m. and found that the
body temperature was 80°F.

The county coroner examined the body at 8 a.m. and found that the body
temperature was 72°.

Assuming that the body temperature was 98° when the person died and that the air
temperature was a

constant 40° all night, what was the approximate time of death? Show all of your
work or explain how

you came up with your solution.

37. Theophylline, an asthma medicine, is to be prepared from an elixir with a
drug concentration of

5 mg/mL and a cherry-flavored syrup that is to be added to hide the taste of the
drug. How much of

each must be used to prepare 100 milliliters of solution with a drug
concentration of 2 mg/mL?

Show all of your work or explain how you came up with your solution.

38. The Lincoln Park Zoo has different admission prices for adults and children.
When Mr. and Mrs.

Weaver went with their five children, the bill was $33. If Mrs. Wong and her
three children got in for

$18.50, then what is the price for an adult’s ticket and what is the price of a
child’s ticket? Show all of

your work or explain how you came up with your solution.

39. The employees from maintenance go for coffee together
every day at 9 a.m. On Monday, Hector paid

$5.45 for three cartons of milk, four cups of coffee, and seven doughnuts. On
Tuesday, Guillermo paid

$3.50 for four milks, two coffees, and eight doughnuts. On Wednesday, Anna paid
$5.15 for two milks,

five coffees, and six doughnuts. On Thursday, Alphonse had to pay for five
milks, two coffees, and nine

doughnuts. How much change did he get back for his $10 bill? Show all of your
work or explain how

you came up with your solution.

40. Farmer Bill is planning to raise wheat and barley. Each acre of wheat yields
a profit of $50, and each

acre of barley yields a profit of $70. To sow the crop, he arranges to borrow a
tractor and a tiller from

his neighbor. The tractor is available for 200 hours and the tiller is available
for 100 hours. Sowing an

acre of barley requires 3 hours of tractor time and 2 hours of tilling. Sowing
an acre of wheat requires

4 hours of tractor time and 1 hour of tilling. How many acres of each crop
should Bill plant to

maximize his profit? Show all of your work or explain how you came up with your
solution.

41. Dr. Pheblitz instructs his nurse to give Mr. Bennett .5 mg of Medex. The
only form of Medex in stock is

a liquid containing .125 mg per 4 mL. How much of this liquid should the nurse
give Mr. Bennett?

Show all of your work or explain how you came up with your solution.

42. A child weighing 33 lb is ordered amoxicillin oral suspension 150 mg by
mouth every 8 hours. Is this a

safe dose for a child of that weight? The label states that children should
receive 20 to 40 mg/kg/day.

Show all of your work or explain how you came up with your solution.|

43. A machined block 3.345 inches thick must be reduced by 18%. What is the new
thickness after the

reduction? Show all of your work or explain how you came up with your solution.

44. A landscape architect plans a circular reflection pool for a park. The area
regulations will not permit a

pool any larger than 254.4696 square meters. What is the largest permissible
diameter that may be used

for this reflection pool? Show all of your work or explain how you came up with
your solution.

45. Rebecca Thomas is paid $285 per week by the Art Lovers of the Lake, plus a
2% commission on all

sales over $500. During a certain week, her total sales were $764. Find her
gross earnings. Show all of

your work or explain how you came up with your solution.

46. Suppose you can rent a $500 television set for $30 a month and after 24
months you own it. Is this a

good idea, or would it be better to charge it on your credit cared and pay off
that credit card account at a

rate of $30 a month? Show all of your work or explain how you came up with your
solution.

47. The diameter of a bolt is .5 inch and the bolt has a tensile strength of
64,000 psi. How much force can

the bolt withstand? Show all of your work or explain how you came up with your
solution.

48. The nineteenth century English mathematician Augustus De Morgan once made
the following

statement: “I was x years old in the year x^{2}.” In what year was De
Morgan born? Show all of your

work or explain how you came up with your solution.

49. You just aced your first math test and you want to celebrate by filling your
bathtub with champagne.

Your bathtub has a rectangular base and vertical sides. It is 7 feet long, 3
feet wide, and 4 feet deep.

How many quarts of champagne will you need to fill your bathtub to a foot below
the top? Show all of

your work or explain how you came up with your solution.

50. The price (per pound) of bananas is linearly related to the mean July
temperature in Quirigua,

Guatemala. When the mean July temperature is 90°F, bananas cost $1.29 per pound.
When the mean

July temperature is 93°F, bananas cost $1.34 per pound. What would be the price
per pound if the mean

temperature was 85°F?

Answer Key

1. approximately 23 weeks

2. I = 7.5t + 115, $212.50 billion dollars

3. scatter plot; y = 289.8x + 1164.364; r = .9836096226; 5511; fairly confident
because r is close to 1

4. answers may vary

5. 24.04 feet per second

6. 16.4 mg/l per hour; after 3 hours

7. increasing on (0, 3), (6, 15); decreasing on (3, 6) and (30, 39); constant on
(15, 30)

8. .4 cm

9. g(x) = 16 - x; x-axis reflection & vertical shift up 16

10. A=36π t^{2}

11. w = .45x

12. gives the number of years since 1970
until the town’s population is P thousand

13. 250 yards long and 70 yards wide

14. 5 mph

15. 180 months or 15 years

16. 229.94 feet

17 Both dividing by a complex number and rationalizing a denominator involve
multiplying both the

numerator and the denominator by the conjugate of the denominator.

18. 3.83 seconds

19. T > 279.57 K

20. approximately .5 second

21. maximum radius 4.480 inches, minimum radius 4.432 inches

22. 1970 (x = 20); 2002 (x≈52); inc from 1959 to 1970, dec from 1970 til now;
the catch was over

2.5 million metric tons from roughly 1955 til 1984

23. approximately 104.5 feet per second

24. y = -2.432x^{3} + 37.317x^{2} - 70.372x +1043.788,1092.421

25. 13

26. 5, May of 1908, 790, February of 1907

27. 5%

28. $228.26, $10,956.48, $1956.48

29. $10,472.80, $121.80

30. approximately 40 parsecs

31. y = 14.33759 - .62561 ln x, where x is the term of the security and y is the
interest rate, 12.642%

32. approximately 48 hours

33. ,2.535micrograms .

34. 5.35%

35. approximately 4.9 seconds

36. 5:20 a.m.

37. 25 mL of elixir and 75 ml of syrup

38. $6.50 for an adult, $4 for a child

39. $3.95

40. 20 acres of wheat and 40 acres of barley for a maximum profit of $3800

41. 16 mL

42. The ordered dose of 450 mg in 24 hours is within the 300 - 600 mg/day safe
range for a child weighing 33 lb.

43. 2.7429 inches

44. 18 meters

45. $290.28

46. The annual interest rate of the rent-to own option is about 40% and the
nominal rate is 22%. If your credit

card has an annual percentage rate lower than 22%, it would be best to use your
credit card.

47. approximately 12, 566 lbs

48. 1806

49. approximately 1885 quarts

50. $1.21