Practice Problems For Final Exam

1. Your car gets 30 miles to the gallon on highway trips. You have $50 and gas costs on average $1.40 per gallon. How far can you afford to drive?
2. In 1990 the cost of a certain computer was $1500, 30% less than the cost in 1980. What was the price of the computer in 1980?
3. Suppose onions at a store in Switzerland are priced at 0.60 francs per kilogram, where one dollar is worth 1.43 francs. What is the price of the onions in dollars per pound? State your answer to the nearest $0.01 per pound. The price for onions in Greece is 129 drachmas per kilogram. Assuming that onions in Switzerland and Greece have the same worth, what is the conversion between francs and drachmas?
4. If an afternoon movie costs $3.50 and a night movie costs $6.00. How much cheaper is the afternoon movie? How much more expensive is the night movie? (express your answers both in absolute terms and as percentages)
5. While out shopping, you notice that at the Gap a certain shirt costs $26.75. At Banana Republic, the same shirt costs $34.25. The Banana Republic shirt is what percentage more than the Gap shirt? If Banana Republic had a 30% sale while the Gap had a 15% sale, then the Banana Republic shirt would be what percentage more or less than the Gap shirt?
6. If enrollment at the university is down 16%, what percent increase is required to return to the original enrollment, before the drop? What percent increase would be required so that enrollment would have increased 5% over the enrollment before the drop?
7. Perform the following conversions
   1. Convert 20 pounds per square inch to grams per square centimeters
   2. Convert 8765 millimeters to kilometers
   3. Convert 40,000 nl to cubic centimeters (Note: 1 ml = 1 $cm^3$)
   4. Convert 64 cubic miles to cubic kilometers.
8. The Great Salt Lake is about 92 miles long and 48 miles wide, with an average depth of 20 feet. The state of Rhode Island has an area of about 1045 square miles. If all of the water from the Salt Lake was spread evenly over Rhode Island, how deep would the water be?
9. Suppose that in 1980 the average rent for a two bedroom apartment was $250, whereas today it would be $600. Which of the following statements are true? If they are false correct the percentage. All answers must be justified by a computation.
   1. The average rental cost increased 140% from 1980 to 1999.
   2. The average rental cost today is 140% of what is was in 1980.
   3. The average rental cost in 1980 is 58.3% of what it is today.
   4. The average rental cost was 58.3% lower in 1980 than today's cost.

10. How much grain can Farmer Bob's silo hold (to the right)? If he wanted to paint the silo, how many gallons of paint would he need assuming a coat thickness of 0.25in (Note: 1 gallon = 231in$^{3}$)? If Farmer Bob's son Bobby were going to paint a scale model which was scaled at 30 to 1, how much paint would he need?

    
11. You are considering opening a savings account. Bank A is offering a 3.7% APR with monthly compounding and bank B is offering 3.6% APR with daily compounding. Which bank should you choose? Justify your answer.
12. You want to save $50,000 over the next 18 years. Assume an APR of 8% with monthly compounding. How much would you have to deposit today (and leave alone) to reach your savings goal? How much would you have to deposit each month to reach your savings goal?
13. You open a savings account with a 4% APR by depositing $100 per month for two years. Then, you get a raise and you increase your monthly payments to $150 per month. You continue to save for 3 more years. How much money do you have in your account after your 5 years of savings?
14. You are saving for retirement. When you retire you want to live off only the interest from your savings. You want to have $40,000 per year to live on. Suppose you will be saving for 40 years. Assume an APR of 8%. How much do you have to save per month to reach your goal? (To make your calculation easier assume simple interest after retirement.)
15. You buy a $135,000 house. You take a mortgage with an APR of 7% for 30 years.
    1. What will your monthly payment be?
    2. As a percentage of the total amount you pay back, how much interest did you pay?
16. You can afford $250 per month for car payments, and you are able to get a loan for 5 years with an APR of 7%.
    1. With no down payment, what is the most expensive car you can afford?
    2. If you had some savings to cover a down payment of 9% of the car price, what is the most expensive car you can afford?
17. If you have an account with an APR of 6% compounded monthly, how long will it take your money to double?
18. Suppose you have an account with an APR of 4% compounded monthly. If you open the account with an initial deposit of $3000 and make no other deposits or withdrawals, how long will it take for you balance to reach $3500?
19. Suppose you have an account with an APR of 5% compounded monthly. You make monthly deposits of $100. How long will it take for you balance to reach $10,000? In absolute and relative terms, how much longer would it take to save the same amount if you were not receiving any interest?
20. Fill in the blanks with ``constant'', ``increasing'', or ``decreasing''. If a quantity is growing linearly the absolute rate of change is , and the relative rate of change is . If a quantity is growing exponentially the absolute rate of change is , and the relative rate of change is .
21. Suppose the population of a country was 3 million people in 1990, and has been growing by 3.2% per year. Create a model, making sure to label your variables. What will the population be in 2025. How long does it take for the population to double?
22. You buy a candle. For each 1.5 hours you burn the candle, it decreases in length by 4.2 cm. You have been burning the candle for 3.75 hours and it is 9.5 cm long. Create a equation to model the burning of the candle. How long was the candle before it burned at all. If you were to let the candle burn continuously from when it was new until it burned out, how long would it burn?
23. Suppose you collected data on the number of students enrolled at a particular school each year. Your independent variable is in years and your dependent variable is enrollment. You plot the data and decide that it is exponential, and you want an equation relating year and enrollment. You make a log plot and find the line of best fit. You find the slope of your log plot to be 0.0107. What can you say about the change in enrollment per year? (Hint: $log(Q) = t(log(1+r)) + log(Q_o)$)
24. You are going to rent a car and you contact two rental companies. Friendly Rentals tell you that they charge $30 per day plus $0.07 per mile. Cheapskate Rentals will charge $20 per day plus $0.30 per mile. For each company, create an equation to model the cost of renting a car from them. Analyze, with an explicit computation, under what circumstances makes one company the better deal over the other.
25. The half life of carbon 14 is 5730 years. Suppose you dig up a bone in a field for which 72% of the carbon 14 has decayed. How old is the bone?

26. You open a small lunch cafe. Every 3 months you record the average number of customers you have for lunch. You do this for just over 2 years, and make a plot of the data. Find the equation of the best fit line. Make a statement about the meaning of the slope and intercept of this line.

    
27. Suppose the population of a town is doubling every 15 years, and in 1980 the population was 30,000. Create a model, making sure to label your variables. What will the population be in 2000? In what year will the population reach 100,000? By what percent does it grow each year?
28. Compute the areas and perimeters of the figures below. What would the new areas be if both figures were scaled by 10?
    
    
29. Suppose you build a scale model of a house which is 25 times smaller than the original. The floor space of your model is 700 square inches. What is the floor space of the real house. The real house has a pool with a volume of 2000 cubic feet. What is the volume of the model pool in cubic inches?