# Two Tough SAT Math Modeling Questions You Should Review

Both questions in this post are official ones sourced from The College Board's question of the day app. The dates referred to in this post will be different if you have a newer version of the app. However, the app just cycles through the same questions, so everything in this post will still be relevant to you. You just won't be able to track down questions according to their date.

#### 1. October 31st, 2015

Scarlett is studying the neurology of mice. One article she reads suggests that the number of neurons a mouse has, $$N(t)$$, after $$t$$ months is modeled by the function $$N(t) = N_0\cdot (1 - r)^{t/\tau}$$, where $$N_0$$ is the initial number of neurons at time $$t = 0, r$$ is a rate where $$0 \lt r \lt 1$$, and $$\tau$$, measured in months, is a constant. If $$\tau$$ is 4 months, which of the following statements would best describe the relationship between the number of neurons and the number of months $$t$$ that have passed since $$N_0$$ was measured?

A) The number of neurons decreases by a fixed amount every 4 months.
B) The number of neurons remaining increases by a fixed amount every 4 months.
C) The number of neurons decreases by an amount proportional to the decrease in the previous 4 months.
D) The number of neurons increases by an amount proportional to the increase in the previous 4 months.

The best way to interpret this problem clearly is to make up some numbers.

Let $$r = 0.50$$.

When $$t = 4$$, $$N(4) = N_0(0.50)^1 = N_0(0.50)$$
When $$t = 8$$, $$N(8) = N_0(0.50)^2 = N_0(0.50)(0.50)$$
When $$t = 12$$, $$N(12) = N_0(0.50)^3 = N_0(0.50)(0.50)(0.50)$$

As you can see, the number of neurons is halved every 4 months. So if we start out with 100 neurons, the number will drop to 50 after four months, and then 25 after another four months.

Answers (B) and (D) are clearly wrong because the number of neurons doesn't increase; it decreases. Answer (A) is wrong because the number of neurons does not decrease by a fixed amount every 4 months.

The government of a particular country is planning to subsidize electric vehicles by lowering their prices in order to encourage consumers to purchase them. Electric vehicles are currently being purchased steadily at a rate of about 287 per month, with an average purchase price of $24,627. For each drop of$1,000 in the average electric vehicle's purchase price, it is expected that the monthly purchase rate will increase by 48. Which of the following best models the monthly electric vehicle revenue, $$R(p)$$, in this country as a function of $$p$$ price reductions of $1,000 each? ($$\text{revenue} = \text{price of product} \cdot \text{ number of products sold}$$) A) $$R(p) = 24,627(287 - 952p)$$ B) $$R(p) = 1,000(24,627 - p)(287 + 48p)$$ C) $$R(p) = (24,627 - p)(287 + p)$$ D) $$R(p) = (24,627 - 1,000p)(287 + 48p)$$ After $$p$$ price reductions of$1,000 each, the average price of an electric vehicle will be $$24,627 - 1,000p$$ and the monthly purchases of electric vehicles will be $$287 + 48p$$. Using the formula given in the question,
$\text{revenue} = \text{price of product} \cdot \text{ number of products sold}$ $R(p) = (24,627 - 1,000p)(287 + 48p)$