SAT Math Multiple Choice Question 476: Answer and Explanation
11. The decline of a certain animal species' population, currently estimated to be 22,000, can be modeled by the quadratic function p(x) = –0.5x2 + 22,000, where x is the number of years after 2015. Based on only this information, and assuming no intervention to change the path of the population, which of the following statements must be true?
- A. This species will be extinct by the end of the year 2225.
- B. The animal population for this species is decreasing at a constant rate.
- C. In approximately 100 years, the animal population for this species will be about half what it was in 2015.
- D. The animal population will increase or decrease from the initial 2015 level, depending on the year after 2015.
Correct Answer: A
Category: Problem Solving and Data Analysis / Scatterplots
Strategic Advice: One of the keys to doing well on Test Day is knowing when (and how) to use your calculator and when it would be quicker to solve something conceptually or by hand. You might try graphing the equation in your calculator, but finding a good viewing window may be very time-consuming. Instead, think about what you know about quadratic functions and how to evaluate them.
Getting to the Answer: Skim through the answer choices to see which ones are easiest to eliminate. The question states that the function is quadratic; therefore, the population cannot be decreasing at a constant rate (or the function would be linear), so eliminate B. A quick examination of the equation tells you that the parabola opens downward (–0.5x2) and its vertex has been shifted up 22,000 units to (0, 22,000). Because x = 0 represents 2015, for all years after 2015 (to the right of 0), the graph will always be decreasing, which means you can eliminate D. The other two choices involve actual numbers, so go back to A. The year 2225 is 210 years after 2015, so the statement translates as "at x = 210, p(x) = 0 (or less theoretically depending on the month of the year)," or more specifically that p(210) = 0. Substitute 210 for x in the equation and see what happens (this is where your calculator is needed): p(210) = –0.5(210)2 + 22,000 = –50. The population can't be negative, but this tells you that by the end of the year 2225, there will be no more of this species, meaning it will be extinct, so (A) is correct.