Math: 9afe2370

ID: 9afe2370

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Exponential modeling 101.

Method 1: The simplified version of this formula, one that applies to savings accounts in real life, is as follows:

amount = starting value * (1.00 +/- rate as a decimal)^time

In the equation given, r is the rate of change, and if the population is decreasing, then r must be negative. However, it cannot be less than -1, since the population at that point would be less than 0, which cannot occur. Thus, -1 < r < 0, and the correct answer is (B).

Method 2: It would take more creativity to create a useful graph than would be worth your time. You could, for instance, model a function y = P(1 + r)^x on Desmos and use sliders for P and r, or you could simply plug in a value for P and then use a slider for r alone. Either way, this question is probably best approached conceptually, since such knowledge can be brought to bear in more challenging ways in other questions.

Method 3: Consider what would happen in each case if the answer choices were correct. Select a number that conforms to the inequality.

A. P = P_0(1 + (-2))^y; a negative value within the group will make the population decrease too much, below 0. If, say, the population were 1000 people and y were something like 1, you can appreciate the point: P = (1000)(-1)^(1). This answer choice cannot be correct.

B. P = P_0(1 + (-0.5))^y; in this case, the population will be half of what it was the previous year, but, crucially, that population can never dip below 0. It is a decreasing population, and the equation makes sense. The correct answer is (B).

C. P = P_0(1 + (0.5))^y; the group will become 1.5, which would represent a 50 percent increase. This is definitely not what you want.

D. P = P_0(1 + (2))^y; the group will now become 3, even greater than what you encountered in the last answer choice. This group would be increasing substantially (by 200 percent) each year.