A laboratory culture begins with bacteria. Every minutes, the population increases by . Which equation defines , where is the estimated population after half-hour intervals and is a whole number greater than and less than ?

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SAT Nonlinear Functions Question 214 | Free SAT Prep | Aniko

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Question 214·Easy·Nonlinear Functions

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A laboratory culture begins with $400$ bacteria. Every $30$ minutes, the population increases by $25\%$ .

Which equation defines $P$ , where $P(n)$ is the estimated population after $n$ half-hour intervals and $n$ is a whole number greater than $1$ and less than $10$ ?

For percent growth problems, first convert the percent change to a decimal and add 1 to get the growth factor (here, $1+0.25=1.25$ ). Recognize that repeated equal percent changes lead to exponential models of the form $P(n)=P_0\cdot(a)^n$ , where $P_0$ is the initial amount and $a$ is the growth factor. Check that the base of the exponent is greater than 1 for growth (less than 1 for decay), and, if needed, quickly verify by plugging in $n=1$ or $n=2$ to see which choice matches the situation.

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Work out just one time step

Start with 400 bacteria. If the population increases by $25\%$ in 30 minutes, what is the population after just one 30-minute interval?

Turn the percent into a multiplier

A $25\%$ increase means you keep $100\%$ of the original and add $25\%$ more. What single number can you multiply 400 by to get that new amount?

Think about repeated multiplication

Every half-hour the population is multiplied by the same factor. When a quantity is multiplied by the same factor over and over, should the factor be in the base or in the exponent?

Desmos Guide

Enter each answer choice as a function

In Desmos, type four functions, one for each answer choice, replacing $n$ with $x$ (for example, use y1 = the right-hand side of choice A, y2 = the right-hand side of choice B, etc.).

Make a quick value table

For each function, click the gear icon and add a table, or use the + button to create a table, so you can see $y$ -values for several $x$ -values like $x=1$ and $x=2$ .

Compare with the actual growth description

Using the problem description, compute the population by hand for the first couple of intervals (for example, after 1 interval and after 2 intervals) and then see which function’s table gives exactly those same values for $x=1$ and $x=2$ . That matching function corresponds to the correct equation.

Step-by-step Explanation

Convert the percent increase to a growth factor

A $25\%$ increase means the new amount is the original plus $25\%$ of the original.

$25\%$ as a decimal is $0.25$ .
After one half-hour, the population becomes:

400 + 0.25\cdot 400 = 1.25\cdot 400

So each 30-minute interval multiplies the population by a factor of $1.25$ .

See the pattern over several intervals

Apply the $1.25$ growth factor repeatedly:

After 1 interval: $400\cdot 1.25$
After 2 intervals: $(400\cdot 1.25)\cdot 1.25 = 400\cdot (1.25)^2$
After 3 intervals: $400\cdot (1.25)^2\cdot 1.25 = 400\cdot (1.25)^3$

Each new half-hour interval adds another factor of $1.25$ as an exponent.

Write the general expression for n intervals and match the choice

Following the pattern, after $n$ half-hour intervals, the population is

P(n) = 400\cdot (1.25)^n.

This matches answer choice A, $P(n) = 400(1.25)^n$ , so choice A is correct.

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Step-by-step Explanation

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