Chapter 3.5, Problem 34E

a.

To determine

To find: the type of $P\left(t\right)$ .

Answer to Problem 34E

Polynomial function

Explanation of Solution

Given information:

  $P\left(t\right)=1000-19.75t+20t^2-\frac{1}{3}{t}^3$

Here P is population of bacteria in a sample and t is number of days.

Calculation:

The given function can be expressed in the form of as shown below.

  $P\left(t\right)=a_1{t}^n+a_2{t}^{n-1}+a_3{t}^{n-3}+\mathrm{.....}+a_n$

Here $a_1,a_2,a_3\mathrm{.....},a_n$ are real numbers.

Thus, the function $P\left(t\right)$ is polynomial.

b.

To determine

To find: t (number of days) when bacteria population increases.

Answer to Problem 34E

39.5

Explanation of Solution

Given information:

  $P\left(t\right)=1000-19.75t+20t^2-\frac{1}{3}{t}^3$

Here P is population of bacteria in a sample and t is number of days.

Calculation:

Plot the graph for $P\left(t\right)$ using graphing utility as follows:

  

The graph shows the maxima $\left(t,P\right)$ at $\left(39.5,10881.593\right)$ . Thus the population increases to 10881 in 39.5 days.

c.

To determine

To find: t (number of days) when bacteria population decreases.

Answer to Problem 34E

0.5

Explanation of Solution

Given information:

  $P\left(t\right)=1000-19.75t+20t^2-\frac{1}{3}{t}^3$

Here P is population of bacteria in a sample and t is number of days.

Calculation:

Plot the graph for $P\left(t\right)$ using graphing utility as follows:

  

You may need to adjust your screen to find the point of minima in the graph.

  

The graph shows the maxima $\left(t,P\right)$ at $\left(0.5,995.083\right)$ . Thus the population decreases to 995 in 0.5 day.