Exponential and Log related word problems:
1) Dana’s mother gave her $175 on her sixteenth birthday. “But you must put it in the bank and leave it there until your eighteenth birthday,” she told Dana. Dana already had $237.54 in her account, which pays 3.25% annual interest, compounded quarterly. What is the minimum amount of money she will have on her eighteenth birthday if she makes no withdrawals before then?
2) An investment counselor advises a client that a safe plan is to invest 30% in bonds and 70% in a low risk stock. The bonds currently have an interest rate of 7% and the stock has a dividend rate of 9%. Write an equation and solve to find how much the client needs to invest to have an annual income of $5,000 from his investment.
3) Barnaby’s grandfather is always complaining that back when he was a kid, he used to be able to buy his girlfriend dinner for only $1.50. If that same dinner that Barnaby’s grandfather purchased for $1.50 sixty years ago now costs $25.25, find the rate in increase annually.
4) At the Write-a-Text Factory, workers spend grueling hours at computer terminals trying to be creative writing textbooks. One day, during a brief bout of boredom, Karen and Carlos decided to play a trick on Darrel as he worked on his document. They implanted a strange code in his computer so that as soon as he had entered 60,000 characters, 10% of his document, starting at the beginning would be deleted every hour. Just as the clock struck 5:00 p.m. and he was anxiously waiting for the whistle to blow telling him he could go home, Darrel typed in his 60,000th character. He left and did not return until 8:00 a.m. the next day. How many characters were left when he returned the next morning?
5) Ever eat a MAGGOT? Guess again! The FDA publishes a list, the Food Defect Action Levels List, which indicates limits for “natural and unavoidable” substances in processed food (Time, October 1990). So, in 100g of mushrooms, for instance, the government allows 20 maggots! The average rich and chunky spaghetti sauce has 350 grams of mushrooms. How many maggots will that contain?
6) If gasoline now costs $1.25 per gallon and is increasing at 5% per year, how long will it be before it costs $2.00 per gallon? Write an equation and solve.
7){a} I would like to have $40,000 in 8 years, and I only have $1,000 now. What interest rate would I need, when compounded yearly, to reach my goal?
{b} Suppose I start with $7,800 and I want to have $18,400 twenty years from now. What interest rate do I need (comp yearly)?
8) An exponential function contains the two points: (3, 12.5) & (4, 11.25). Is this an increasing or decreasing exponential? If the horizontal asymptote for this function is the line y=10 and there is no horizontal shift, find the equation of the exponential function that would go through the given points.
9) The economy has worsened to the point that the merchants in downtown Hollywood cannot afford to replace the light bulbs when they burn out. On average about thirteen percent of the light bulbs burn out every month. Assuming there are now about one million outside store lights in Hollywood, how long will it take until there are only 100,000 bulbs lit? How about only 1 bulb still lit?
10) A rule of thumb used by car dealers is that the trade-in value of a car decreases by 20% each year. Suppose the initial value of the car is $23,500. How much is the car worth in four years? In how many years will the trade-in value be $6,000? If a car is really 2.7 years old now, what was its trade-in value when it was new?
11) The “half-life” (the time it takes to reduce the initial amount to half of the initial amount) of uranium is 1000 years. If 50 grams of uranium is sealed in a box, how much is left after 10,000 years? How long will it take to reduce to 1% of the original amount? How long will it take until all of the original mass of uranium is gone?