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f4f27229-89b0-45f9-a957-f871d7a68cc7.html
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<h4>Activity</h4>
<p>The value of some cell phones changes exponentially after initial release. Here are graphs showing the depreciation of two phones 1, 2, and 3 years after they were released.</p>
<p>Phone A</p>
<p><img alt="Graph of an exponential function, origin O. time (years) and value (dollars)." src="https://k12.openstax.org/contents/raise/resources/ed255e46d77de5a8f2270113e81764f77b8780eb"></p>
<p>Phone B</p>
<p><img alt="Graph of an exponential function, origin O. time (years) and value (dollars). " src="https://k12.openstax.org/contents/raise/resources/25bf06446555406d7db5441aef3160353edd937f"> </p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3d468a0b-200d-4fb2-b7b8-4cfee6bbb375" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li> Which phone is more expensive to buy when it is first released? </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> Phone A costs $1,000 and Phone B costs $840, so Phone A is more expensive. </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="4f101a5b-20b5-4dac-958b-9b25f23e753b" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li> Which graph depicts the value of the cell phone falling in value more quickly? Explain how you know. </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>Phone A is depreciating more quickly than phone B. This was determined by comparing the value of the phones each year. For instance, after one year, phone A was only worth $600 compared to phone B's value of $630. And, again, after three years, phone A was valued at $216 compared to phone B's value of $354.38.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3bb9c5b7-0aef-4ece-bb21-17271f17303e" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>How does the value of phone A change with every passing year? </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> Phone A loses $400 in the first year, which is \(\frac25\) (or 0.4 or 40%) of the original value. It loses $240 in the second year, which is \(\frac25\) (or 0.4 or 40%) of $600. </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="e3d6c1e3-2b80-4f29-8c9c-76ae62d057f0" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>How does the value of phone B change with every passing year? </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> Phone B loses $210 in the first year, which is \(\frac14\) (or 0.25 or 25%) of the original value, and $157.50 in the second year, which is \(\frac14\) of $630.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ae8606bb-0c51-4707-9e19-d10a61506f02" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li> If the phones continue to depreciate by the same factor each year, what will the value of each phone be 4 years after its initial release?
<ol class="os-raise-noindent" type="a">
<li>Enter your answer for Phone A.</li>
</ol>
</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> \(\frac{3}{5} \cdot $216= $129.60\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3ee9fa05-46b7-4a9b-b06b-92769f2a5e5e" 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>Enter your answer for Phone B.</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>\(\frac{3}{4} \cdot ($354.38) \approx $265.79\) </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="3b70c454-2ef1-4dbb-b89c-cb913b0092fc" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>For each cell phone, write an equation that relates the value of the phone in dollars to the years since release, \(t\). Use \(v\) for the value of Phone A and \(w\) for the value of Phone B.
<ol class="os-raise-noindent" type="a">
<li>Enter your answer for Phone A.</li>
</ol>
</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> \( v=1,000 \cdot (\frac{3}{5})^t\) </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="fc5a7ce1-6ca5-4643-950a-b58732756d8b" 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>Enter your answer for Phone B.</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> \( w=840 \cdot (\frac{3}{4})^t\) </p>
</div>
</div>
<!--Interaction End -->
<br>
<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-input" data-button-text="Solution" data-content-id="516284cb-8fd5-41b0-9da5-1fb4cf196a81" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>When given data, it is not always clear how to best model it. In this case, we were told the value of the cell phones was changing exponentially. Suppose, however, we were instead just given the initial values of the cell phones when released and the values after each of the first three years.</p>
<ol class="os-raise-noindent">
<li>Use technology to compute the best fit line for each cell phone. Round any numbers to the nearest dollar.
<ol class="os-raise-noindent" type="a">
<li>Enter your answer for Phone A.</li>
</ol>
</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> \(v=933−259t\) </p>
</div>
</div>
<!--End interaction-->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="9712c068-c01b-4fa4-abfe-0e883761278e" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li>Enter your answer for Phone B.</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>\(
w=816−161t\) </p>
</div>
</div>
<!--End interaction-->
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="908748e4-4caf-4353-8aa8-2cf90194284c" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Explain why, in this situation, an exponential model might be more appropriate than the linear model you just created. </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 linear model would quickly lead to negative phone values. For example, Phone A would cost about –$103 four years after its initial release. It makes more sense that the phone would lose a fraction of its value each year. </p>
</div>
</div>
<!--End interaction-->