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Find the annual multiplier for the following.
A yearly increase of 5% due to inflation.
A two–bedroom house in Nashville is worth $110,000. If it appreciates at a rate of 2.5% each year, what will it be worth in 10 years?
When will it be worth $200,000?
In Homewood, houses are depreciating at a rate of 5% each year. If a house is worth $182,500 now, how much would it be worth two years from now?
For the following situation, identify the multiplier, the initial value, and the time. Remember that the time must be in the same units as the multiplier. (Example: 3% raise per quarter for two years: multiplier = 1.03, time = 8 quarters)
A 1970 comic book has appreciated at 10% per year, and originally sold for 35 cents.
Sean wants to know its value in the year 2000.
Suppose you invest $300 at 6% annual interest, compounded yearly. How much will you have after 7 years?
Find the multiplier and time for the following.
A yearly increase of 1.23% in population.
Kristin's grandparents started a savings account for her when she was born. They invested $100 in an account that pays 8% interest compounded annually.
Write an equation to express the amount of money in the account on Kristin's xth birthday.
How much is in the account on her 16th birthday?
What are the domain and range for the equation that you wrote in part (a)?
Suppose you invest $1000 at 12% annual interest, compounded yearly. How much will you have after 2 years?
Elle has moved to Hawksbluff for one year and wants to join a health club. She has narrowed her choices to two places: Thigh Hopes and ABSolutely fABulous. Thigh Hopes charges a fee of $95 to join and a monthly fee of $15. ABSolutely fABulous charges a fee of $125 to join and $12 per month. Which club should she join?
One thousand dollars is invested at 12% interest annually. Determine how much the investment is worth after a year.
For each of the problems below, find the initial value.
Five years from now, a bond that appreciates at 4% per year will be worth $146.
Even though A = P e^{rt} applies only to continuously compounded interest, it gives a pretty good approximation for standard exponential growth in many circumstances. Consider this example: If food prices increase steadily at 7% per year over a 10 year period, how much will a $20 bag of groceries cost in 10 year?
a) Answer the question using the standard exponential function C = km^{t}, where C = future cost, m = yearly multiplier, and t = number of years elapsed.
b) Answer the question using A = P e^{rt}.
c) Is the answer to part (b) a suitable approximation to the answer to part (a)?
For the following situation, identify the multiplier, the initial value, and the time. Remember that the time must be in the same units as the multiplier. (Example: 3% raise per quarter for two years: multiplier = 1.03, time = 8 quarters)
Inflation is at the rate of 7% per year. Today Janelle's favorite bread costs $1.79. What would it have cost ten years ago?
The gross national product (GNP) was 1.665 10^{12} dollars in 1960. If the GNP increased at the rate of 3.17% per year until 1989:
a) what was the GNP in 1989?
b) write an equation to represent the GNP t years after 1960, assuming that the rate of growth remained constant.
Suppose you invest $1000 at 8% annual interest, compounded yearly. How much will you have after 2 years?
Macario's salary is increasing by 5% each year. His rent is increasing by 8% each year. Currently, 20% of Macario's salary goes to pay his rent. Assuming that Macario does not move or change jobs, what percent of his income will go to pay rent in 10 years? Think about this problem and if you see a way to solve it, do it. If not, complete the four parts below.
If x represents Macario's current salary write an expression to represent his salary ten years from now.
Use x in an expression to represent Macario's rent now.
Use what you wrote in part (b) to write an expression for the rent ten years from now.
Now use the expressions that you wrote in parts (a) and (c) to write a ratio that will help you answer the question.
You want to invest some amount of money at 7.5% interest, compounded yearly. If you want to have $800 after 10 years, how much do you need to invest?
For each of the problems below, find the initial value.
Seventeen years from now, Ms. Speedi's car, which is depreciating at 20% per year, will be worth $500.
You have $5000 in an account that pays 12% annual interest. Compute the amount in the account at the end of one year if:
the interest is compounded annually (once)
A rule–of–thumb used by car dealers is that the trade–in value of a car decreases by 20% each year.
Explain how the phrase "decreases by 20% each year " tells you that the trade–
in value varies exponentially with time (i.e., can be represented by an exponential function).
Suppose the initial value of the car is $23500. Write an equation expressing the trade–in value of your car as a function of the number of years from the present.
How much is the car worth in four years?
In how many years will the trade–in value be $6000?
If the car is really 2.7 years old now, what was its trade–in value when it was new?