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6fbb32e9-f411-4ac6-97b1-d323c504e0be.html
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<h4>Activity</h4>
<p>A person took some medicine but does not remember how much. Concerned that she took too much, she has a blood test
every hour for several hours.</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">\(t\), time (hours)</th>
<th scope="col">\(m\), medicine (mg)</th>
</tr>
</thead>
<tbody>
<tr>
<td>
–3
</td>
<td> </td>
</tr>
<tr>
<td>
–1
</td>
<td> </td>
</tr>
<tr>
<td>
0
</td>
<td>
100
</td>
</tr>
<tr>
<td>
1
</td>
<td>
50
</td>
</tr>
<tr>
<td>
2
</td>
<td>
25
</td>
</tr>
</tbody>
</table>
<br>
<p>Answer the following questions. Double click on mathematical expressions/equations to enlarge.
</p>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="8dee9721-b2c9-430f-8853-ad513788661f" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>Time \(t\) is measured in hours since the blood test, and amount of medicine in her body, \(m\), is
measured in milligrams. What is the decay factor? That is, what is \(b\) in an equation of the form
\(m=a\cdot b^t\)?</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> \(b=\frac12\)</p>
</div>
</div>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="eea90097-e49a-496d-b71f-f4f93c74d8fa" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li> What is \(a\)?</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>\(a=100\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="2647ff5a-cf0f-41c0-bb43-79f5b08dde16" 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="3">
<li>Find the amounts of medicine in the patient’s body when \(t\) is –1.</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>When \(t\) is –1, \(m\) is 200.</p>
</div>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="89e682b4-b099-486a-9270-cbb7965899ee" 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="4">
<li>Find the amounts of medicine in the patient’s body when \(t\) is –3.</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>When \(t\) is –3, \(m\) is 800.</p>
</div>
</div>
<!--Interaction End -->
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="23abee67-c162-431f-aed0-e59dc0580d32" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li>What does \(t=0\) mean in this context?</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> \(t=0\) means the time the doctor did the first blood test.</p>
</div>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="18bd1a08-62fc-412d-b68b-4ce4b79bee1a" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>What does \(t=-3\) mean in this context?</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> \(t=-3\) means 3 hours before the doctor did the first test.</p>
</div>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="8795c0f1-1934-42c0-8831-8573d01030fd" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="7">
<li>The medicine was taken when \(t\) is –5. Assuming the person did not have any of the medication in
her body beforehand, how much medicine did the patient take?</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>3,200 mg. Each hour, this is cut in half. After being halved 5 times, it leaves 100 mg, the amount of medicine in
the person’s body when \(t=0\).</p>
</div>
</div>
<!--Interaction End -->
<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="8">
<li>Create a graph by plotting the points whose coordinates are in the table from question 1 above. Make sure to
draw and label tick marks on the axes.</li>
</ol>
<p><img height="296" src="https://k12.openstax.org/contents/raise/resources/bd9e16f57c87cb863d69fea1d8f03d9f67d48df6" width="300">
</p>
</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 answer:</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="220" role="presentation" src="https://k12.openstax.org/contents/raise/resources/87c0825791421cf128a96a1b9eabd7e5cd7ccd97" width="300"></p>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="5a89aa91-ac46-43e1-aee6-c7000b31c629" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="9">
<li>Based on your graph, when do you think the patient will have: 500 mg of medicine remaining in her body</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>Between 2 and 3 hours before the patient first had her blood tested</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="521b36a3-ce92-4c88-a86f-aad8418328ec" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="10">
<li>Based on your graph, when do you think the patient will have: No medicine remaining in her body</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:
<br> Your answer may vary, but here are some samples:
</p>
<ul>
<li>Sometime after 10 hours. The amount of medicine falls below 0.1 mg after about 10 hours and might be too small
to be detectable after that point.</li>
<li> Eventually, all of the medicine will be out of the patient’s body (but the mathematical model for the
amount of medicine does not reach 0).</li>
</ul>
</div>
</div>
<!--Interaction End -->
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="61ef699f-d746-4ab1-9447-3eceb0c3b48b" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem" data-content-id="a1f54a9b-cea1-45c5-ac43-2ac7329663a6" data-problem-type="dropdown" data-solution="close to 1" data-solution-options='["close to 0", "close to 1","much larger than 1"]' data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem-content">
<h4>Extending Your Thinking</h4>
<br>
<p>Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger
than 1.</p>
<ol class="os-raise-noindent">
<li>\(\frac1{1-2^{-10}}\)<br>
</li>
</ol>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is: close to 1</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is: close to 1</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="5ca31138-9e4f-4b4d-ad63-509dbdb6a8ef" data-problem-type="dropdown" data-solution="close to 1" data-solution-options='["close to 0", "close to 1","much larger than 1"]' data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem-content">
<br>
<p>Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger
than 1.</p>
<ol class="os-raise-noindent" start="2">
<li>\(\frac{2^{10}}{2^{10}+1}\)<br>
</li>
</ol>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is: close to 1</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is: close to 1</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="edc064ea-76ed-49fe-aa30-d41afe493426" data-problem-type="dropdown" data-solution="close to 0" data-solution-options='["close to 0", "close to 1","much larger than 1"]' data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem-content">
<br>
<p>Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger
than 1.</p>
<ol class="os-raise-noindent" start="3">
<li>\(\frac{2^{-10}}{2^{10}+1}\)<br>
</li>
</ol>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is: close to 0</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is: close to 0</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="a715a50b-aebc-4bb7-9e97-346d824a0243" data-problem-type="dropdown" data-solution="close to 0" data-solution-options='["close to 0", "close to 1","much larger than 1"]' data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem-content">
<br>
<p>Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger
than 1.</p>
<ol class="os-raise-noindent" start="4">
<li>\(\frac{1-2^{-10}}{2^{10}}\)<br>
</li>
</ol>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is: close to 0</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is: close to 0</p>
</div>
</div>
<div class="os-raise-ib-pset-problem" data-content-id="fff56ade-4ac8-4bb1-95e8-015e427ddb17" data-problem-type="dropdown" data-solution="much larger than 1" data-solution-options='["close to 0", "close to 1","much larger than 1"]' data-wait-for-event="RFM1">
<div class="os-raise-ib-pset-problem-content">
<br>
<p>Without evaluating them, describe each of the following quantities as close to 0, close to 1, or much larger
than 1.</p>
<ol class="os-raise-noindent" start="5">
<li>\(\frac{1+2^{10}}{2^{-10}}\)<br>
</li>
</ol>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is: much larger than 1</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! The answer is: much larger than 1</p>
</div>
</div>
<!--Do not edit below this line-->
<div class="os-raise-ib-pset-correct-response"> </div>
<div class="os-raise-ib-pset-attempts-exhausted-response"> </div>
</div>
<!--End interaction-->