SAT Nonlinear Functions Question 227 | Free SAT Prep | Aniko

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Question 227·Medium·Nonlinear Functions

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A laboratory begins an experiment with 400 bacteria. Every 5 hours, the number of bacteria triples.

Which equation gives $b(t)$ , the number of bacteria $t$ hours after the experiment begins?

For exponential growth word problems, first recognize the structure: $\text{amount} = (\text{initial}) \times (\text{growth factor})^{\text{number of periods}}$ . From the wording, extract the initial value, the growth factor per period (e.g., triples means factor 3), and the length of one “period” (here, 5 hours). Then express the exponent as the number of those periods in $t$ units of time (usually $t/\text{period length}$ ). Finally, plug these into the exponential form and be careful not to mix up $t/5$ with $5t$ , and not to use $1/3$ when the quantity is increasing by a factor of 3.

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Think about the type of model

Is the bacteria count changing by a constant amount each hour, or by being multiplied by the same factor over equal time intervals? What type of function models that?

Identify the key components

Pick out the initial number of bacteria and the factor by which the bacteria count changes every 5 hours. How do those appear in an exponential function?

Focus on the exponent

The exponent should represent how many times the population is multiplied by 3. How many 5-hour periods are there in $t$ hours?

Check special times

Whichever equation you choose, plug in $t=0$ and $t=5$ . It should give 400 at $t=0$ and 1200 at $t=5$ (since the population triples after 5 hours).

Desmos Guide

Enter each option as a function

In Desmos, type each option as a separate function, for example:

y = 400*(3)^(5x)
y = 400*(3)^(x/5)
y = 400*(1/3)^(x/5)
y = 400/(3^(x/5)) Treat $x$ in Desmos as the time $t$ in hours.

Check the initial value at t = 0

Use the table feature or tap on $x=0$ for each graph. The correct model must give $y=400$ when $x=0$ , since the experiment starts with 400 bacteria. Eliminate any function that does not show 400 at $x=0$ .

Check the value after 5 hours

Now look at $x=5$ for the remaining functions. The bacteria should triple after 5 hours, so the correct function must give $y=1200$ when $x=5$ . The option whose graph shows this value is the correct model.

Step-by-step Explanation

Recognize exponential growth structure

The bacteria count is multiplying by the same factor (tripling) over equal time intervals (every 5 hours), so this is an exponential growth situation.

The general form is:

$\text{amount}(t) = (\text{initial amount}) \times (\text{growth factor})^{\text{number of periods}}$ .

Identify the initial amount and growth factor

From the problem:

Initial amount = 400 bacteria, so the function should start with a factor of $400$ .
The bacteria triples every 5 hours, so the growth factor per period is $3$ (not $1/3$ ).

Find the exponent: number of 5-hour periods in t hours

The exponent must count how many times the population triples.

One “period” is 5 hours.
In $t$ hours, the number of 5-hour periods is $t/5$ .

So the exponent on 3 should be $t/5$ (not $5t$ ).

Write the function for b(t)

Combine the pieces:

Initial amount: 400
Growth factor: 3
Exponent (number of 5-hour periods): $t/5$

So the function is

b(t) = 400(3)^{t/5}

which is the equation that correctly models the bacteria count after $t$ hours.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation