Faaiz M. answered • 07/15/20
TJHSST Straight-A Graduate with Perfect ACT Score
Hi Park!
This is an exponential decay problem. We know this because the number of pathogens, h, decreases by a certain amount of itself (1/3) every 2 hours. We can use our basic exponential decay formula, which says a(1-b)x. Here, "a" is our initial value, "b" is the fraction by which it declines, and "x" is the time that has passed. So "n" takes the place of a, and "1/3" takes the place of "b." But there's a twist on x! It doesn't decline by 1/3 every ONE hour, it declines by 1/3 every TWO hours! To understand how that will change the answer, let's examine why the formula is what it is. In the original a(1-b)x, every time x increases by 1, you multiply the initial value by 1-b. Say your initial value is 8, and you decrease by 1/2 of the value each turn. So we start with 8, then we multiply that by (1 - 1/2), which is (1/2), once. We get 4, which looks right. But the next time we do it, we make that x value 2. So now we multiply 8, our initial value, by (1 - 1/2) which is (1/2), twice! In other words, (1/2)2 * 8, which is 2. That looks right! So let's apply this to your question. If the pathogen number was reduced by ⅓ of its value, or in other words, was multiplied by (2/3) every ONE hour, we’d have n(1 - ⅓)h. But we don’t! It multiplies by that value every TWO hours. So we want the multiplication to happen half as often as that formula would suggest. So we say n(⅔)h/2, which is D.
