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409c44ff-bc3f-46f8-9944-a0680ca12c58.html
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<h4>Activity</h4>
<p>A patient who is diabetic receives 100 micrograms of insulin. The graph shows the amount of insulin, in micrograms,
remaining in his bloodstream over time, in minutes.</p>
<p><img alt="Graph of an exponential function, origin O. time (minutes) insulin (mg)." src="https://k12.openstax.org/contents/raise/resources/13ce593bdb8d48b45fef1b04c12b0cd0087995c1"></p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="183a9495-d9b6-433c-810b-6c713ba0a1bd" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>
Scientists have found that the amount of insulin in a patient’s body changes exponentially. How can you
check if the graph supports the scientists’ claim?
</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
If the insulin decays exponentially, that means that after each minute, the same fraction of insulin will decay and
remain. So one way to check if this prediction matches the data is to see what fraction of insulin decays (or
remains) after the first minute, then after the second minute, and so on. If these fractions are the same, then the
graph supports the scientists’ claim.
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1c78a80d-40f8-4ddb-8c7a-346118b7dd93" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Answer the following questions:
</li>
<ol class="os-raise-noindent" type="a">
<li>
How much insulin broke down in the first minute?
</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
10 micrograms broke down.
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ad928825-1b24-47fe-b621-fa646c84772a" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What fraction of the original insulin is that?</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
\(\frac{1}{10}\) of the initial injection
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="85eddfb3-ff6e-4b87-a52a-a9fd7a0cd8fd" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li> Answer the following questions.
</li>
<ol class="os-raise-noindent" type="a">
<li>
How much insulin broke down in the second minute? </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
9 micrograms broke down.
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="307c5fa8-a3e0-499d-a1fa-56a172cd50fa" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What fraction is that of the amount one minute earlier?</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
\(\frac{1}{10}\) of the amount one minute earlier
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="53c44400-1a22-47f0-82b4-e5923402e26c" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>
What fraction of insulin remains in the bloodstream for each minute that passes? Be prepared to show your
reasoning.
</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(\frac{9}{10}\). For example: In the next minute, another \(\frac{1}{10}\) of insulin broke down. Each time,
\(\frac{9}{10}\) of the amount remains. </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="5">
<li>
Complete the table to show the predicted amount of insulin 4 and 5 minutes after injection.</li>
</ol>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">Time after Injection (Minutes)</th>
<td>
0
</td>
<td>
1
</td>
<td>
2
</td>
<td>
3
</td>
<td>
4
</td>
<td>
5
</td>
</tr>
<tr>
<th scope="row">Insulin in the Bloodstream (Micrograms)</th>
<td>
100
</td>
<td>
90
</td>
<td>
81
</td>
<td>
72.9
</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.
</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answers:</p>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
<tr>
<th scope="row">Time after Injection (Minutes)</th>
<td>
0
</td>
<td>
1
</td>
<td>
2
</td>
<td>
3
</td>
<td>
4
</td>
<td>
5
</td>
</tr>
<tr>
<th scope="row">Insulin in the Bloodstream (Micrograms)</th>
<td>
100
</td>
<td>
90
</td>
<td>
81
</td>
<td>
72.9
</td>
<td>
65.61
</td>
<td>
59.05
</td>
</tr>
</tbody>
</table>
</div>
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="36341f6b-6367-4a1a-9c66-c93fb0652a4d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>Answer the following questions.
</li>
<ol class="os-raise-noindent" type="a">
<li>
Describe how you would find how many micrograms of insulin remain in his bloodstream after 10 minutes.
</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Multiply 100 by \(\frac{9}{10}\) ten times: \(100 \cdot (\frac{9}{10})^{10}\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="a06f26c2-b24d-44d8-901e-330ac2ee011d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<ol class="os-raise-noindent" start="2" type="a">
<li>
<p>
After \(m\) minutes? </p>
</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> For example: Multiply 100 by \(\frac{9}{10}\) \(m\) times: \(100 \cdot (\frac{9}{10})^m\).</p>
</div>
</div>
<!--Interaction End -->
<br>
<h4> Video: Analyzing Exponential Graphs and Writing Equations</h4>
<p>Watch the following video to learn more about analyzing exponential graphs and using the graphs to write an equation.
</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/fe068a2a74f0c4e7b754654213f3c60db3177152">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/300ba95bc7f806826d3d86dd18e3878d5a8689fc" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/fe068a2a74f0c4e7b754654213f3c60db3177152
</video></div>
</div>