Assignment:
Part I : Life Span The table below gives the life expectancy for the people in the United State for the birth years 1910-1998.
a. Find the logarithmic function that models these data, with x equal to 0 in 1900.
b. Find the quadratic function that is the best fit for the date. Round the quadratic coefficient to five decimal places.
c. Graph each of these functions on the same axes with the data points to determine visually which function is the best model for the data for the years 1910-1998.
d. Evaluate both models for the birth year 2010. Which model is better for prediction of life span after 2010?
|
Birth Year
|
Life Span
(years)
|
Birth Year
|
Life Span
(years)
|
Birth Year
|
Life Span
(years)
|
|
1910
|
50.0
|
1981
|
74.2
|
1990
|
75.4
|
|
1920
|
54.1
|
1982
|
74.5
|
1991
|
75.5
|
|
1930
|
59.7
|
1983
|
74.6
|
1992
|
75.5
|
|
1940
|
62.9
|
1984
|
74.7
|
1993
|
75.5
|
|
1950
|
68.2
|
1985
|
74.7
|
1994
|
75.7
|
|
1960
|
69.7
|
1986
|
74.8
|
1995
|
75.8
|
|
1970
|
70.8
|
1987
|
75.0
|
1996
|
76.1
|
|
1975
|
72.6
|
1988
|
74.9
|
1997
|
76.5
|
|
1980
|
73.7
|
1989
|
75.2
|
1998
|
76.7
|
Part II : Cell Phones The following table gives the number of millions of U.S. cellular telephone subscribers.
a. Create a scatter plot for the data with x equal to the number of years from 1985. Does it appear that the data could be modeled with a quadratic function?
b. Find the quadratic function that is the best fit for these data, with x equal to the number of years from 1985 and y equal to the number of subscribers in millions?
c. Use the model to estimate the number in 2005.
d. What part of the U.S. population does this estimate equal?
|
Year
|
Subscribers(millions)
|
Year
|
Subscribers(millions)
|
|
1985
|
0.340
|
1994
|
24.134
|
|
1986
|
0.682
|
1995
|
33.786
|
|
1987
|
1.231
|
1996
|
44.043
|
|
1988
|
2.069
|
1997
|
55.312
|
|
1989
|
3.509
|
1998
|
69.209
|
|
1990
|
5.283
|
1999
|
86.047
|
|
1991
|
7.557
|
2000
|
107.478
|
|
1992
|
11.033
|
2001
|
128.375
|
|
1993
|
16.009
|
2002
|
140.767
|
Part III : World Population One projection of the world population by the United Nation for selected years (a low projection scenario) is given in the table below.
|
Year
|
Projected Population(million)
|
Year
|
Projected Population(million)
|
|
1995
|
5666
|
2075
|
6402
|
|
2000
|
6028
|
2100
|
5153
|
|
2025
|
7275
|
2125
|
4074
|
|
2050
|
7343
|
2150
|
3236
|
a. Find a quadratic function that fits these data, using the number of the years after 1990 as the input.
b. Find the positive x-intercept of this graph, to the nearest year.
c. When can we be certain that this model no longer applies?
Part IV : Classroom Size The date in the table below give the number of students per teacher for selected years between 1960 and 1998.
|
Year
|
Students per Teacher |
Year
|
Students per Teacher
|
|
1960
|
25.8
|
1992
|
17.4
|
|
1965
|
24.7
|
1993
|
17.4
|
|
1970
|
22.3
|
1994
|
17.3
|
|
1975
|
20.4
|
1995
|
17.3
|
|
1980
|
18.7
|
1996
|
17.1
|
|
1985
|
17.9
|
1997
|
17.0
|
|
1990
|
17.2
|
1998
|
17.2
|
|
1995
|
17.3
|
|
|
a. Find the power function that is the best fit for the data, using as input the number of years after 1950.
b. According to the unrounded model, how many students per teacher were there in 2000?
c. Is this function increasing or decreasing during this time period?
d. What does the model predict will happen to the number of student per teacher as time goes on?
Part V : College Tuition New parent want to put a lump sum into a money market fund to provide $300,000 in 18 years, to help pay for college tuition for their child. If the fund average 10% per year compounded monthly, how many should they invest?
Part VI : Business Sale A man can sell his Thrifty Electronics business for $800,000 cash or for $100,000 plus $122,000 at the end of each year for 9 years.
a. Find the present value of the annuity that is offered if money is worth 10% compounded annually.
b. If he takes the $800,000, spends $100,000 of it, and invests the rest in a 9-year annuity at 10% compounded annually, what size annuity payment will he receive at the end of each year?
c. Which is better, taking the $100,000 and the annuity of taking the cash settlement? Discuss the advantage of your choice.
Part VII : Loan Repayment A loan of $10,000 is to be amortized with quarterly payments over 4 years. If the interest on the loan is 8% per year, paid on the unpaid balance,
a. What is the interest rate charged each quarter on the unpaid balance?
b. How many payments are made to repay the loan?
c. What payment is required each quarterly to amortize the loan?
Part VIII : Home Mortgage A couple who wants to purchase a home with a price of $350,000 has $100,000 for a down payment. If they can get a 30-year mortgage at 6% per year on the unpaid balance,
a. What will be their monthly payment?
b. What is the total amount they will pay before they own the house outright?
c. How much interest will they pay over the life of the loan?
Part IX : Sexually Active Boys The percent of boys between ages 15 and 20 that been sexually active at some time (the cumulative percent) can be modeled by the logistic function
Y = 89.786 / 1 + 4.6531e - 0.8256x
Where t is the number of years after age 15.
a. Graph this function for 0 < x < 5.
b. What does the model estimate the cumulative percent to be for boys whose age is 16?
c. What cumulative percent does the model estimate for boys of age 21, if it is valid after age 20?
d. What is the limiting value implied by this model?
Part X : Spread of Disease An employee brings a contagious disease to an office with 150 employees. The number of employees infected by the disease t days after the employees are first exposed to it is given by
N = 100 / 1 + 79e¯0.9t
Use graphical or numerical methods to find the number of days until 99 employees have been infected.
Provide complete and step by step solution for the question and show calculations and use formulas.