The amount of a certain drug (in milligrams) remaining in a patient’s bloodstream hours after injection is modeled by the function According to the model, about how many hours after the injection will only milligrams of the drug remain in the bloodstream?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 107 | Free SAT Prep | Aniko

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Question 107·Medium·Nonlinear Functions

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The amount of a certain drug (in milligrams) remaining in a patient’s bloodstream $t$ hours after injection is modeled by the function

M(t)=200e^{-0.3t}.

According to the model, about how many hours after the injection will only $50$ milligrams of the drug remain in the bloodstream?

For exponential decay equations on the SAT, first set the given formula equal to the target amount, then isolate the exponential term by dividing both sides. Next, take the natural log (ln) of both sides to bring down the exponent and solve the resulting linear equation for the variable. Use your calculator only at the end to evaluate the logarithm and pick the nearest answer choice, which saves time and reduces rounding errors.

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Write the equation you need to solve

You know $M(t) = 200e^{-0.3t}$ and you want the time when $M(t)$ is $50$ . What equation does that give you?

Isolate the exponential expression

Once you have $200e^{-0.3t}$ equal to a number, what can you divide both sides by to get $e^{-0.3t}$ by itself?

Get the exponent out of the exponential

When you have an equation like $e^{\text{(expression)}} = \text{number}$ , what operation can you use on both sides to bring the exponent down?

Use your calculator wisely

After solving algebraically for $t$ in terms of a natural log, use your calculator to approximate and then pick the closest answer choice.

Desmos Guide

Enter the decay model

In Desmos, type y = 200e^(-0.3x) to graph the amount of drug over time, where $x$ represents hours.

Graph the target amount

Add a second equation y = 50 to represent the horizontal line where the drug amount is 50 milligrams.

Find the intersection

Tap or click on the point where the curve y = 200e^(-0.3x) intersects the line y = 50. Read off the $x$ -coordinate of this point; that $x$ -value is the time (in hours) when 50 milligrams remain.

Step-by-step Explanation

Translate the question into an equation

The function $M(t)=200e^{-0.3t}$ gives the amount of drug (in milligrams) after $t$ hours.

We want the time $t$ when only $50$ milligrams remain, so set

200e^{-0.3t} = 50.

Isolate the exponential term

Solve for the exponential part $e^{-0.3t}$ by dividing both sides by $200$ :

\begin{aligned} \frac{200e^{-0.3t}}{200} &= \frac{50}{200} \\ e^{-0.3t} &= \frac{50}{200} = 0.25 \end{aligned}

Use natural logarithms to solve for t

To get the exponent out of $e^{-0.3t}$ , take the natural log of both sides:

\begin{aligned} \ln\big(e^{-0.3t}\big) &= \ln(0.25) \\ -0.3t &= \ln(0.25) \\ t &= \frac{\ln(0.25)}{-0.3} \end{aligned}

Since $0.25=\tfrac{1}{4}$ , $\ln(0.25)=-\ln(4)$ , so

t = \frac{\ln(4)}{0.3}.

Approximate the value and choose the closest answer

Use a calculator to approximate:

$\ln(4) \approx 1.386$
So

t = \frac{\ln(4)}{0.3} \approx \frac{1.386}{0.3} \approx 4.62.

This is about $4.6$ hours, so the correct answer choice is 4.6.

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