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Mathematics - Exponential & Logarithmic Functions

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Please resend the solutions below.

Exercise 3.4 (from posting 267232, was missing the table, so was unable to get the solution)

23. 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
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

Exercise 2.5 (from posting 266708, did not receive the solutions)
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

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?

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?

Exercise 3.6 (From posting 267226/267232, did not receive solutions)

9. 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?

17. 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.

21. 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?

23. 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?

11. 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?

23. 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.

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Solution Summary

The exponential and logarithmic functions are examined. The limiting value implied by the model are determined. A complete, neat and step-by-step solutions are provided in the attached file.

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