SAT Math Multiple Choice Question 769: Answer and Explanation
4. P(t) = 250(2.4)t
The formula above shows the relationship between the population, P, of a certain mushroom species on a one-acre plot of land as a function of t, the number of weeks that have passed since the mushrooms were first introduced on the plot.
By what percent should we expect the mushroom population to increase between the start of week 6 and the start of week 7?
- A. 40%
- B. 140%
- C. 240%
- D. 480%
Correct Answer: B
Advanced Mathematics (exponential functions) MEDIUM-HARD
The fastest way to answer this question is to notice that in exponential growth and decay functions in which the exponent is the time variable, t, the base of the exponential (in this case 2.4) must represent 1 + r, where r is the rate of change per time unit. This means that the rate of weekly increase (recall that t is measured in weeks) must be 2.4 - 1 = 1.4 = 140%.
Another way to solve the problem is to calculate the populations at the specified times and then calculate the percent change. At the beginning of the sixth week, 5 weeks have passed, and so the population is 250(2.4)5 = 19,906. At the beginning of the seventh week, the population is 250(2.4)6 = 47,776. To calculate the percent change, we find the difference and divide by the initial amount: (47,776 - 19,906)/19,906 = 1.40 = 140%.