Which equation below would allow a person to determine his savings account balance after 12 years, on an initial investment of $500 that pays a 4% annual interest? Assume he does not touch this money within the 12 years.
A. y = 500(1.04)12
Correct! In the general form showing exponential growth, y = a(1 + r) to the x power, a represents the initial investment, $500, r represents the growth rate, 4%, and x represents the time intervals that have passed, 12 years.
B. y = 1200(1.04)12
Incorrect. In this exponential equation, the initial investment is $1200, which is incorrect, because in this scenario, the initial investment was only $500.
C. y = 500(1.12)12
Incorrect. In this exponential equation, the growth rate is 12%, which is incorrect, because in this scenario, the annual interest was 4%.
D. There is not enough information to formulate an equation.
Incorrect. There is sufficient information to formulate an exponential equation since you are given the initial investment, $500, the growth rate, 4%, and the amount of time intervals, 12 years.
1990, there were 285 cell phone subscribers in the small town of Phonyville. The number of subscribers increased by 65% per year every year after 1990. Assuming the rate of increase stayed the same, which exponential equation below would allow you to find the number of cell phone subscribers in Phonyville in 2010?
A. y = 285(1.65)2010
Incorrect. In this exponential equation, the number of time intervals, 2010, is incorrect. From 1990 to 2010, there are 20 annual increases of 65% per year.
B. y = 285(65)20
Incorrect. In this exponential equation, the annual growth of 65%, or 0.65, needs to be added to 1, y = a(1+r) x because each year’s growth is 65% more than the previous year’s quantity… not just a growth factor of 65.
C. y = 285(1.65)20
Correct! In the general form showing exponential growth, y = a(1+r) x, a represents the initial # of cell phone subscribers, 285, r represents the growth rate, 65% or .65, and x represents the time intervals that have passed, 20 years.
D. y = 285 + 20(65)
Incorrect. This linear model shows the initial number of subscribers at 285 but shows a constant growth of 65 subscribers each year for 20 years.
Each year the local country club sponsors a tennis tournament. The tournament starts with 256 participants. During each round, half of the players are eliminated. Which exponential equation below could be used to determine how many rounds will be played before a winner is declared?
A. y = 256(1 − 5)x
Incorrect. This exponential equation shows the correct initial amount, 256 players, and it shows the correct decay rate, 50% or 0.5, but in order to answer the question and find ‘x’, the number of rounds played until a winner is declared, you must substitute ‘1’ for ‘y’ in the equation.
B. 1 = 256 − .5x
Incorrect. This is a linear equation which shows that ½ of a person is eliminated each round. That doesn’t make sense.
C. 1 = 256(1 +.5)x
Incorrect. This exponential equation shows a growth rate of 50% per round but this is an elimination tournament so the number of players will decrease each time.
D. 1 = 256(1 − .5)x
Correct! This exponential equation shows the correct initial amount, 256 players, the correct decay rate, 50%, and will allow one to determine x, the number of rounds necessary to have one winner.
The brewing pot temperature of coffee is approximately 180 degrees F. The equation, y = 180(.987)x, represents the cooling temperatures, in degrees F, of a freshly brewed cup of coffee x minutes after it has been poured from the brewing pot into a serving cup.
In 1992, McDonald’s was sued for serving its coffee at a temperature that caused a woman to be severely burned when the coffee spilled. In the testimony, it was stated that if the coffee had been served at 155 degrees, then serious burns would have been avoided.
Which equation below could a restaurant owner use to determine how long waiters should wait, after pouring coffee from the brewing pot, before serving it to customers at a temperature of 155 degrees?
A. 155=180(.987)x
Correct! Solving this exponential equation would allow you to find x, the number of minutes of time that must lapse until temperature reaches 155 degrees.
B. 180 = 155(.987)x
Incorrect. In this exponential equation, the initial temperature of the coffee is 155 degrees but the scenario states that the initial temperature is 180 degrees.
C. y = 180 − 155x
Incorrect. This linear equation shows the coffee temperature cooling at a constant rate of 155 degrees per minute.
D. y = 180(.987)155
Incorrect. This exponential equation shows the coffee cooling for 155 minutes… over two hours!