SAT Nonlinear Functions Question 96 | Free SAT Prep | Aniko

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Question 96·Hard·Nonlinear Functions

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$P(t)=\dfrac{1{,}200}{1+4e^{-0.3t}}$

A biologist models the number of bacteria in a culture $t$ minutes after a nutrient is introduced with the function above.

According to the model, to the nearest whole minute, how long after the nutrient is introduced will the population first reach 900 bacteria?

For equations involving exponential or logistic models like this, set the function equal to the target output (here, 900), then solve for the input by: 1) clearing any fractions, 2) isolating the exponential term, and 3) taking natural logs on both sides to bring down the exponent and solve for the variable. On multiple-choice SAT questions, you can also quickly test the answer choices by plugging each time into the given function with your calculator and seeing which one gives a value closest to the target; choose the smallest time that reaches or is closest to that value if the question asks for the first time something happens.

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Use the model equation directly

You are looking for the time $t$ when the population is 900, and you are given $P(t)$ . Start by setting $P(t)$ equal to 900 and writing an equation involving $t$ .

Clear the denominator

Once you have $900 = \dfrac{1200}{1+4e^{-0.3t}}$ , eliminate the fraction by multiplying both sides by the denominator, then simplify and isolate the exponential term $e^{-0.3t}$ .

Deal with the exponent

When you have an equation of the form $e^{\text{(something)}} = \text{number}$ , what operation lets you bring the exponent down? Use that operation here to solve for $t$ .

Check with answer choices or a calculator

After you solve for $t$ algebraically, approximate its decimal value with your calculator and then round to the nearest whole minute to see which answer choice it matches.

Desmos Guide

Graph the population function

In Desmos, enter the function as y = 1200/(1 + 4e^(-0.3x)) . Make sure you are using x for the input (time in minutes).

Find when the population is 900

Add a horizontal line y = 900 . Use the intersection tool (or tap/click the intersection point) between the curve and the line. Note the x-coordinate of this intersection (the time in minutes) and round it to the nearest whole number to match an answer choice.

Step-by-step Explanation

Set up the equation for 900 bacteria

We are told that the population is modeled by

P(t) = \frac{1200}{1 + 4e^{-0.3t}}

We want the time $t$ when the population first reaches 900 bacteria, so set $P(t)$ equal to 900:

900 = \frac{1200}{1 + 4e^{-0.3t}}.

Now we will solve this equation for $t$ .

Clear the fraction and isolate the exponential

Start by multiplying both sides by the denominator to eliminate the fraction:

900(1 + 4e^{-0.3t}) = 1200.

Divide both sides by 300 to simplify the numbers:

3(1 + 4e^{-0.3t}) = 4.

Distribute the 3:

3 + 12e^{-0.3t} = 4.

Subtract 3 from both sides:

12e^{-0.3t} = 1.

Finally, divide both sides by 12:

e^{-0.3t} = \frac{1}{12}.

Now the exponential term is isolated.

Use natural logarithms to solve for t

To get $t$ out of the exponent, take the natural logarithm (ln) of both sides:

\ln\bigl(e^{-0.3t}\bigr) = \ln\left(\frac{1}{12}\right).

Using the rule $\ln(e^x) = x$ , the left side simplifies to $-0.3t$ :

-0.3t = \ln\left(\frac{1}{12}\right).

Note that $\ln\left(\dfrac{1}{12}\right) = -\ln(12)$ , so

-0.3t = -\ln(12).

Divide both sides by $-0.3$ to solve for $t$ :

t = \frac{\ln(12)}{0.3}.

This is the exact value of the time in minutes.

Approximate and match to the nearest minute

Use a calculator to find $\ln(12)$ and then divide by $0.3$ :

$\ln(12) \approx 2.485$ .
$t \approx \frac{2.485}{0.3} \approx 8.28.$

So the time is about $8.28$ minutes. To the nearest whole minute, this is **8 minutes**, which corresponds to choice **B) 8 minutes**.

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation