MAT 1101 Final Exam A Spring 2000

Name_____________________

There is space on this test to show your work and in some cases partial credit will be given. Make sure that your work is coherent and that you cross out anything that is wrong.

  1. (20 POINTS) The number of homicides in South Park in the four years since Stan Cartman moved to town is given in the table below.

t (years)

1

2

3

4

homicides

2

22

58

225

    1. Find the least squares linear model P(t) = P0 + mt for this data set and its average error

      2.  

       

      P(t) = 70.5t - 99.5

      Average error =_38.93

    2. Find the least squares quadratic model P(t) = P0 + at + bt2 for this data set and its average error.

      3.  

       

       

      P(t) = 36.75t2 - 113.25t + 84.25

      Average error =12.86

    3. Find the least squares natural growth model P(t) = P0at for this data set and its average error.

      4.  

       

       

      P(t) = .625389(4.54408t)

      Average error =21.32

    4. Choose the model above which you think is the best and use it to predict the number of homicides in South Park ten years (t=10) after Stan first moved into South Park. Give a short reason for the choice of your model.

 

 

 

 

 

Answer: The quadratic model, since it has the lowest average error. Ten years from now this model predicts 2627 homicides
  1. You are the owner of the Tasteless Record Corporation (TRC). At a price of $7.00 per CD your corporation can sell 500,000 CD’s of musical sludge a week. For each $.10 increase in price the number of CD’s sold each week decreases by 8000.
    1. Express the number N of CD’s sold each week as a function of the price x.

      2.  

       

       

      N= -80000x+1060000

    2. Express your corporation’s weekly revenue, R, as a function of the price x. (Hint: Revenue equals number of CD’s sold times price.)

      3.  

       

       

      R= x(-80000x+1060000)

    3. Assuming that each week your corporation pays a band $500 to record enough musical sludge to fill a CD and that the manufacturing cost of each CD is $.15, use your answer to (a) to express the cost of producing N CD’s as a function of the price x.

      4.  

       

       

      C= 500+.15(-80000x+1060000)

    4. Find the price x your greedy corporation should charge in order to maximize its profits. (Hint: Profit equals revenue minus cost.)

 

 

 

x= $6.70

  1. You have $10,000 to invest. The First National Bank will pay 5% annual interest each year. The Second National Band will pay 6% annual interest the first year and 4% the second. Compute how much you have in your account after two years and circle which account is the better investment.

    2.  

     

     

     

    First National Bank: $11025 - This bank is the better investment.

    Second National Bank: $11024

     

  2. The population of Villa Mathematica was 200 thousand in 1990. Write a formula P(t) = ... giving the population of Villa Mathematica t years after 1990 if:
    1. The population increases by 7 thousand every 2 years.

      2.  

       

       

       

      Answer: P(t) = 200000+3500t

       

    2. The population decreases by 4 thousand every nine months.

      3.  

       

       

       

       

      Answer: P(t) = 200000-5333t

    3. The population increases by 5% each year.

      4.  

       

       

       

      Answer: P(t) = 200000(1.05)t

       

    4. The population increases by 5% every two years.

      5.  

       

       

       

      Answer: P(t) = 200000(1.05)t/2

       

    5. The population decreases by 5% every two years.

 

 

 

 

Answer: P(t) = 200000(.95)t/2

 

 

 

 

 

 

 

 

 
  1. On January 1, 2000, you deposit $8000 in an account paying 7% annual interest. On January 1 of each subsequent year, you deposit an additional $1000 in this account. How much money will be in this account on January 1, 2008, after that day’s deposit has been made?

    2.  

     

     

     

     

     

    Answer:$24005.29 .

     

  2. On the moon you jump upward from ground level with an initial velocity of 400 ft/second. Your height y after t seconds is given by y(t) = -4t2+v0t+y0.
    1. What is your maximal height above the ground?

      2. Answer: 10000 feet

    2. At what time do you reach this maximal height?

      3.  

       

       

      Answer: 50 seconds

    3. At what time do you return to ground level?

 

 

 

Answer: 100 seconds

  1. The deer population in the state of Georgia was 25 thousand in 1950 and grows naturally.
    1. Write a formula giving the population P(t), t years after 1950, if the number of deer doubles every 8 years.

      2.  

       

      P(t) = 25000(2)^(t/8)

       

    2. How long (accurate to the nearest month) does/did/will it take this population to grow to 1 million?

 

 

Answer:42 years, 7 months

8. You want to construct a rectangular box with a square x by x base and height y with a volume of 1000 cubic inches. First use the fixed volume of 1000 cubic inches to write a relation between x and y. Next write a formula A=… for the box’s surface area in terms in terms of x and y. Then write a function A(x)=… for the box’s surface area in terms of x. Finally find the dimensions which minimize the box’s surface area and the minimal surface area.

 

 

 

  1. relation:x2y = 1000

    2.  

     

     

     

     

  2. formula A= 2x2+4xy

    3.  

     

     

     

     

  3. function A(x)= 2x2+4x(1000/x2)

    4.  

     

     

     

     

  4. dimensions 10 by 10 by 10

    5.  

     

     

     

  5. Minimal surface area 600 square feet

(9) Upon graduation from college you receive two job offers. Exploitation Incorporated offers you a starting salary of $40,000 per year with a promise of a $4,000 raises every year. Math Incorporated offers you a starting salary of $35,000 per year with a promise of 8% raises every year.

  1. Write a formula SE(t)=… for the Exploitation Incorporated salary after t years.

    2.  

     

     

    SE(t)= 40000+4000t

  2. Write a formula SM(t)=… for the Math Incorporated salary after t years.

    3.  

     

     

    SM(t)= 35000(1.08)^t

  3. How many years will it take for the Math Incorporated salary to overtake the Exploitation Incorporated salary?

 

 

 

Answer: 11.95 years

 

10) You have $10,000 to invest. The bank of South Park pays 7.2% annual interest ( with yearly compounding ) while the Springfield bank pays 7% annual interest with daily compounding.

  1. How much money will you have in one year if you deposit the $10,000 in the South Park bank?

    2.  

     

     

    Answer: $10720

  2. How much money will you have in one year if you deposit the $10,000 in the Springfield bank?

 

 

 

Answer: $10725.01

11) The amount A(t) of atmospheric pollutants in South Park grows naturally and is tripling every 5 years.

  1. If the initial amount is 10 pu (pollutant units), write a formula giving the amount A(t) present after t years

    2.  

     

     

    A(t) = 10(3)^(t/5) .

  2. What will be the amount of pollutants present in the city after 5 years?

    3.  

     

     

    Answer 30 pu

  3. If it will be dangerous to stay in South Park when the amount of pollutants reaches 100 pu how long will this take?

 

 

 

Answer: 10.48 years