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<div class="exercise-statement">
<p>
<strong>Zfinal.</strong>
</p>
<p> For each of the following scenarios, provide a reasonable IVP model. Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t" alt="t" title="t" data-latex="t"/> represent time, let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?z" alt="z" title="z" data-latex="z"/> represent the value given in the scenario, and use uppercase letters for positive constants. Then label each term and initial value of each IVP to describe what it represents from the scenario. </p>
<ul>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. </li>
<li> An object is released from rest from above the ground. Let z measure its upward velocity at a given time. Assume quadratic air resistance. </li>
<li> Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time. </li>
<li> A solution of salt water is pumped into a tank of initially fresh water, while mixed water is pumped out at the same rate. Let z measure the mass of salt in the tank at a given time. </li>
</ul>
<p> Then, show that your model is equivalent to one of the following IVP models by relabeling constants and using algebra as needed. (It's possible that more than one IVP may be used for a scenario. If so, choose any of them.) </p>
<ol type="a">
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" alt="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" title="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" data-latex="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-E z - {z'} = -A E \hspace{2em}z(0)= V" alt="-E z - {z'} = -A E \hspace{2em}z(0)= V" title="-E z - {z'} = -A E \hspace{2em}z(0)= V" data-latex="-E z - {z'} = -A E \hspace{2em}z(0)= V"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" alt="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" title="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" data-latex="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" alt="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" title="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" data-latex="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" alt="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" title="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" data-latex="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0"/>
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -X z - A {z'} + S \hspace{2em}z(0)= D" alt="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" title="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" data-latex="0 = -X z - A {z'} + S \hspace{2em}z(0)= D"/>
</li>
</ol>
</div>
</mattextxml>
<mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>Zfinal.</strong>
</p>
<p> For each of the following scenarios, provide a reasonable IVP model. Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?t" alt="t" title="t" data-latex="t"> represent time, let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?z" alt="z" title="z" data-latex="z"> represent the value given in the scenario, and use uppercase letters for positive constants. Then label each term and initial value of each IVP to describe what it represents from the scenario. </p>
<ul>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. </li>
<li> An object is released from rest from above the ground. Let z measure its upward velocity at a given time. Assume quadratic air resistance. </li>
<li> Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time. </li>
<li> A solution of salt water is pumped into a tank of initially fresh water, while mixed water is pumped out at the same rate. Let z measure the mass of salt in the tank at a given time. </li>
</ul>
<p> Then, show that your model is equivalent to one of the following IVP models by relabeling constants and using algebra as needed. (It's possible that more than one IVP may be used for a scenario. If so, choose any of them.) </p>
<ol type="a">
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-W%20z%5E%7B2%7D%20+%20S%20X%20+%20S%20%7Bz'%7D%20=%200%20%5Chspace%7B2em%7Dz(0)=0" alt="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" title="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" data-latex="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-E%20z%20-%20%7Bz'%7D%20=%20-A%20E%20%5Chspace%7B2em%7Dz(0)=%20V" alt="-E z - {z'} = -A E \hspace{2em}z(0)= V" title="-E z - {z'} = -A E \hspace{2em}z(0)= V" data-latex="-E z - {z'} = -A E \hspace{2em}z(0)= V">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-C%20E%20+%20%7Bz'%7D%20=%20-%5Cfrac%7BC%20z%7D%7BN%7D%20%5Chspace%7B2em%7Dz(0)=0" alt="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" title="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" data-latex="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-K%20z%20+%20N%20%5Cdelta%5Cleft(-M%20+%20t%5Cright)%20+%20%7Bz'%7D%20=%20-D%20z%5E%7B2%7D%20%5Chspace%7B2em%7Dz(0)=%20V" alt="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" title="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" data-latex="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?Q%20z%20+%20N%20%7Bz'%7D%20+%20V%20%5Cdelta%5Cleft(-J%20+%20t%5Cright)%20=%20-L%20%7Bz''%7D%20%5Chspace%7B2em%7Dz(0)=-%20R%20,z'(0)=0" alt="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" title="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" data-latex="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0">
</li>
<li>
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-X%20z%20-%20A%20%7Bz'%7D%20+%20S%20%5Chspace%7B2em%7Dz(0)=%20D" alt="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" title="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" data-latex="0 = -X z - A {z'} + S \hspace{2em}z(0)= D">
</li>
</ol>
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<div class="exercise-answer">
<h4>Partial Answer:</h4>
<ul>
<li> An object is released from rest from above the ground. Let z measure its upward velocity at a given time. Assume quadratic air resistance. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" alt="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" title="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" data-latex="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0"/></li></ul></li>
<li> A solution of salt water is pumped into a tank of initially fresh water, while mixed water is pumped out at the same rate. Let z measure the mass of salt in the tank at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" alt="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" title="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" data-latex="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0"/></li></ul></li>
<li> Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" alt="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" title="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" data-latex="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V"/></li></ul></li>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" alt="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" title="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" data-latex="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0"/></li></ul></li>
<li> A bowl of hot soup sits in a colder room. Assume the room's temperature is kept constant, and let z measure the temperature of the soup over time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-E z - {z'} = -A E \hspace{2em}z(0)= V" alt="-E z - {z'} = -A E \hspace{2em}z(0)= V" title="-E z - {z'} = -A E \hspace{2em}z(0)= V" data-latex="-E z - {z'} = -A E \hspace{2em}z(0)= V"/></li></ul></li>
<li> A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0 = -X z - A {z'} + S \hspace{2em}z(0)= D" alt="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" title="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" data-latex="0 = -X z - A {z'} + S \hspace{2em}z(0)= D"/></li></ul></li>
</ul>
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<mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<ul>
<li> An object is released from rest from above the ground. Let z measure its upward velocity at a given time. Assume quadratic air resistance. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-W%20z%5E%7B2%7D%20+%20S%20X%20+%20S%20%7Bz'%7D%20=%200%20%5Chspace%7B2em%7Dz(0)=0" alt="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" title="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0" data-latex="-W z^{2} + S X + S {z'} = 0 \hspace{2em}z(0)=0"></li></ul>
</li>
<li> A solution of salt water is pumped into a tank of initially fresh water, while mixed water is pumped out at the same rate. Let z measure the mass of salt in the tank at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-C%20E%20+%20%7Bz'%7D%20=%20-%5Cfrac%7BC%20z%7D%7BN%7D%20%5Chspace%7B2em%7Dz(0)=0" alt="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" title="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0" data-latex="-C E + {z'} = -\frac{C z}{N} \hspace{2em}z(0)=0"></li></ul>
</li>
<li> Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-K%20z%20+%20N%20%5Cdelta%5Cleft(-M%20+%20t%5Cright)%20+%20%7Bz'%7D%20=%20-D%20z%5E%7B2%7D%20%5Chspace%7B2em%7Dz(0)=%20V" alt="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" title="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V" data-latex="-K z + N \delta\left(-M + t\right) + {z'} = -D z^{2} \hspace{2em}z(0)= V"></li></ul>
</li>
<li> A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. After some time, a hammer instantly strikes the mass inward. Assume the presence of friction. Let z measure the outward displacement of the mass from its natural position on the spring. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?Q%20z%20+%20N%20%7Bz'%7D%20+%20V%20%5Cdelta%5Cleft(-J%20+%20t%5Cright)%20=%20-L%20%7Bz''%7D%20%5Chspace%7B2em%7Dz(0)=-%20R%20,z'(0)=0" alt="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" title="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0" data-latex="Q z + N {z'} + V \delta\left(-J + t\right) = -L {z''} \hspace{2em}z(0)=- R ,z'(0)=0"></li></ul>
</li>
<li> A bowl of hot soup sits in a colder room. Assume the room's temperature is kept constant, and let z measure the temperature of the soup over time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-E%20z%20-%20%7Bz'%7D%20=%20-A%20E%20%5Chspace%7B2em%7Dz(0)=%20V" alt="-E z - {z'} = -A E \hspace{2em}z(0)= V" title="-E z - {z'} = -A E \hspace{2em}z(0)= V" data-latex="-E z - {z'} = -A E \hspace{2em}z(0)= V"></li></ul>
</li>
<li> A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time. <ul><li><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?0%20=%20-X%20z%20-%20A%20%7Bz'%7D%20+%20S%20%5Chspace%7B2em%7Dz(0)=%20D" alt="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" title="0 = -X z - A {z'} + S \hspace{2em}z(0)= D" data-latex="0 = -X z - A {z'} + S \hspace{2em}z(0)= D"></li></ul>
</li>
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