5873e1a1-ee03-467d-9ff9-1f93ba3ca1f0.html

<p><strong><em>Students will complete the following questions to practice the skills they have learned in this
  lesson.</em></strong></p>
<ol class="os-raise-noindent">
<li>For an experiment, a scientist designs a can, 20 cm in height, that holds water. A tube is installed at the bottom
of the can allowing water to drain out.</li>
</ol>
<p>At the beginning of the experiment, the can is full. Every minute after the start of the experiment, \(\frac23\) of
the water is drained.</p>
<p>The height of the water, \(h\), in cm, is a function \(f\) of time \(t\)&nbsp;in minutes since the beginning of the
experiment, \(h=f(t)\). Which is an expression for \(f(t)\)?</p>
<ol class="os-raise-noindent"></ol>

<ul>
<li>\(
f(t)=20+ \frac{1}{3}t\) </li>
<li>\(
f(t)=20(\frac{1}{3})^t\)</li>
<li> \(f(t)=20(\frac{2}{3})^t\) </li>
<li> \(f(t)=20+\frac{2}{3}t\) </li>
</ul>
<p> <strong>Answer: </strong> \(
f(t)=20(\frac{1}{3})^t\) </p>
<ol class="os-raise-noindent" start="2">
<li>A bacteria population is 10,000. It triples each day.

The bacteria population, \(b\), is a function of the number of days, \(d\)<em>.</em> Which equation relates \(b\)
and \(d\)?</li>
</ol>
<ul>
<li>\(
b=10000 \cdot 3^d\) </li>
<li> \(b=10000 \cdot 3d\) </li>
<li> \(b=10000(\frac13)^d\) </li>
<li> \(b=10000+3^d\) </li>
</ul>
<p> <strong>Answer:</strong> \(b=10000 \cdot 3^d\) </p>
<ol class="os-raise-noindent" start="3">
<li> The area, \(a\), covered by a city is 20 square miles. The area grows by a factor of 1.1 each year, \(t\), since
it was 20 square miles. Which equation expresses \(a\) in terms of \(t\)? </li>
</ol>
<ul>
<li> \(a=20+1.1^t\) </li>
<li> \(a=20 \cdot 1.1t\) </li>
<li>\(
a=20 \cdot 1.1^t\) </li>
<li> \(a=20-1.1^t\) </li>
</ul>


<p> <strong>Answer</strong>:\( a=20 \cdot 1.1^t\)</p>
<ol class="os-raise-noindent" start="4">
<li>TThe graph below models the water draining out of a tube during an experiment represented by the function, \(f(t) = 20(1/3)^t\). What does the value \((2,
\frac{20}{9})\) mean in the context of the problem? <br><br><img alt class="img-fluid atto_image_button_text-bottom"
  height="594" role="presentation"
  src="https://k12.openstax.org/contents/raise/resources/58c7b5ead8561c6b71a8fcb3563b7d584d51cf0b" width="300">
</li>
</ol>
<ul>
<li> After 2 minutes, the height of the water is \(\frac{20}{9}\) centimeters less than at the start. </li>
<li> After \(\frac{20}{9}\) minutes, the height of the water is 2 centimeters. </li>
<li> After 2 minutes, the height of the water is \(\frac{20}{9}\) centimeters. </li>
<li> After \(\frac{20}{9}\) minutes, the height of the water is 2 centimeters less than at the start. </li>
</ul>

<p> <strong>Answer: </strong> After 2 minutes, the height of the water is \(\frac{20}{9}\) centimeters. </p>


<ol class="os-raise-noindent" start="5">
<li>A scientist measures the height, \(h\), of a tree each month, and \(m\)&nbsp;is the number of months since the
scientist first measured the height of the tree. Is the height, \(h\),&nbsp;a function of the month, \(m\)?</li>
</ol>
<p> <strong>Answer: </strong>Yes</p>
<ol class="os-raise-noindent" start="6">
<li>Cary started with 3 goldfish in her tank, and the population doubled every month. Which is an equation that
represents the population, \(P\), of goldfish after \(m\) months? </li>
</ol>
<ul>
<li>\(
P=3+2m\) </li>
<li> \(P=3(2m)\) </li>
<li> \(P=3 \cdot 2^m\) </li>
<li> \(P=2 \cdot 3^m\) </li>
</ul>

<p> <strong>Answer:</strong> \(P=3 \cdot 2^m\) </p>
<p>Use this equation for problems 7–10: 
\(f(t)=50 \cdot 2^t\)</p>
<ol class="os-raise-noindent" start="7">
<li>What is the initial value?</li>
</ol>

<p> <strong>Answer: </strong>The initial value is 50. When
\(t=0\), \(f(t)=50 \cdot 2^0=f(t)=(50)(1)=50\).</p>
<ol class="os-raise-noindent" start="8">
<li>What is the growth factor?</li>
</ol>

<p> <strong>Answer: </strong>The initial value is 2. The growth factor refers to the constant multiplier that
determines how the function grows or decays as the independent variable changes. In this equation, the growth factor
is 2.</p>
<ol class="os-raise-noindent" start="9">
<li>What is the independent variable?</li>
</ol>

<p> <strong>Answer:</strong> \(t\), The independent variable is the variable that is manipulated or controlled and is
represented by \(t\) in this equation.</p>
<ol class="os-raise-noindent" start="10">
<li>What is the dependent variable?</li>
</ol>

<p> <strong>Answer:</strong>
\(f(t)\), The dependent variable is the variable that depends on or is influenced by the independent variable. In this
equation, the dependent variable is \(f(t)\), which represents the output or value of the function.</p>
<ol class="os-raise-noindent" start="11">
    <li>Which equation is most appropriate for modeling this data?</li>
    </ol>
<table class="os-raise-horizontaltable">
<thead></thead>
<tbody>
  <tr>
    <th scope="row">x</th>
    <td>1</td>
    <td>2</td>
    <td>3</td>
    <td>4</td>
    <td>5</td>
    <td>6</td>
  </tr>
  <tr>
    <th scope="row">y</th>
    <td>50</td>
    <td>100</td>
    <td>200</td>
    <td>400</td>
    <td>800</td>
    <td>1600</td>
  </tr>
</tbody>
</table>
<br>
<ul>
<li>\(y=25 \cdot (2x)^2\)</li>
<li>\(y=25 \cdot 2^x\)</li>
<li>\(y=25 \cdot x^2\) </li>
<li>\(y=50 \cdot 2^{\frac1x}\)<br>
</li>
</ul>