Chapter 5.5, Problem 72PPS

a.

To determine

To solve: the value of the polynomial equation.

Answer to Problem 72PPS

  $t=0,-35,15$ .

Explanation of Solution

Given:

The number of cells of a virus can be modeled by $P\left(t\right)=-0.012t^3+6.3t+8000$ .

Calculation:

According to the question, Therefore,

  $\begin{array}{l}P\left(t\right)=-0.012t^3-0.24t^2+6.3t+8000\left\{\text{given}\right\}\\ 8000=-0.012t^3-0.24t^2+6.3t+8000\left\{\text{substitutes for }P\left(t\right)\right\}\\ 0.012t^3+0.24t^2-6.3t=0\{\text{subtract 8000 from both sides}}\\ 12t^3+240t^2-6300t=0\{\text{multiply 1000 from both sides}}\\ 12t\left(t^2+20t-525\right)=0\{\text{taking }12t\text{ as common factor}}\mathrm{.....}\text{{1}}\end{array}$

Now, to solve $\left(t^2+20t-525\right)=0$

  $\begin{array}{l}\left(t^2+20t-525\right)=0\\ \left(t^2+20t+100-100-525\right)=0\left\{expand\right\}\\ \left({\left(t+10\right)}^2-625\right)=0\text{{as }\left(a^2+2ab+b^2\right)={\left(a+b\right)}^2\}\\ {\left(t+10\right)}^2-{\left(25\right)}^2=0\text{{simplify}\}\\ \left(t+10+25\right)\left(t+10-25\right)=0\text{{as }\left(a^2-b^2\right)=\left(a+b\right)\left(a-b\right)\}\\ \left(t+35\right)\left(t-15\right)=0\text{{simplify}\}\end{array}$

From {1},

  $\begin{array}{l}12t\left(t^2+20t-525\right)=0\left\{from1\right\}\\ 12t\left(t+35\right)\left(t-15\right)=0\left\{abovesolved\right\}\end{array}$

Either $12t=0$ or $\left(t+35\right)=0$ or $\left(t-15\right)=0$

When $12t=0\Rightarrow t=0$

When $\left(t+35\right)=0\Rightarrow t=-35$

When $\left(t+15\right)=0\Rightarrow t=15$

Hence, the value of the polynomial equation is $t=0,-35,15$ .

b.

To determine

To find: the method by which the polynomial factor is obtained.

Answer to Problem 72PPS

Completing the square method to solve the equation.

Explanation of Solution

Given:

The number of cells of a virus can be modeled by $P\left(t\right)=-0.012t^3+6.3t+8000$ .

Calculation:

The polynomial factor is obtained by following steps:

1. Substitutes $P\left(t\right)=8000$
2. Subtract 8000 and then multiply 1000 from both the sides.
3. After that remove the factor of $12t$ and make it a a trinomial.
4. And solve the trinomial by using the method of completing the square.
5. Further, the all factor equates to 0. And solve the required value

c.

To determine

To check: that for any value of t is reasonable and t is unreasonable

Answer to Problem 72PPS

  $t=0,15$ is reasonable and $t-35$ is unreasonable.

Explanation of Solution

Given:

The number of cells of a virus can be modeled by $P\left(t\right)=-0.012t^3+6.3t+8000$ .

Calculation:

From part (a), the value of $t=0,-35,15$ .

As it is known that the time is never negative, therefore

the value of $t=0,15$ is reasonable and is $t-35$ is unreasonable.

Hence, the value of $t=0,15$ is reasonable and $t-35$ is unreasonable.

d.

To determine

To graph: the function for $0\le t\le 20$ .

Answer to Problem 72PPS

The a graphing utility is shown in below figure.

Explanation of Solution

Given:

The number of cells of a virus can be modeled by $P\left(t\right)=-0.012t^3+6.3t+8000$ .

Calculation:

As the given polynomial function is $P\left(t\right)=-0.012t^3+6.3t+8000$ .

Now, graph the function for $t\in \left[0,20\right]$ using a graphing utility.