Question

10. A bacteria culture is started with 200 bacteria. After 4 hours, the population has grown to 769 bacteria. If the population grows exponentially according to the formula Pt=P0(1+r)t (a) Find the growth rate. Round your answer to the nearest tenth of a percent.r = %(b) If this trend continues, how many bacteria will there be in one day? bacteria(c) How long will it take for this culture to triple in size? Round your answer to the nearest tenth of an hour. hours

Answer 1

Given:

A bacteria culture is started with 200 bacteria.

After 4 hours, the population has grown to 769 bacteria.

the population grows exponentially according to the formula:

`P(t)=P_0(1+r)^t`

(a) Find the growth rate.

so, when:

`\begin{gathered} P_0=200,t=4,P(t)=769 \\ 769=200(1+r)^4 \end{gathered}`

Solve for r:

`\begin{gathered} (769)/(200)=(1+r)^4 \\ 3.845=(1+r)^4 \\ \sqrt[4]{3.845}=1+r \\ 1+r=1.4 \\ r=1.4-1=0.4 \end{gathered}`

So, the value of r = 0.4 = 40%

The growth rate = r = 40%

(b) If this trend continues, how many bacteria will there be in one day?

For one day, t = 24 hours

so,

`\begin{gathered} P_t=200\cdot(1+0.4)^(24) \\ P_t=200\cdot1.4^(24)=642,840 \end{gathered}`

Bacteria = 642,840

(c) How long will it take for this culture to triple in size?

So,

`\begin{gathered} P_t=3\cdot200=600 \\ 600=200\cdot(1+0.4)^t \end{gathered}`

Solve for t:

`\begin{gathered} (600)/(200)=1.4^t \\ 3=1.4^t \\ \ln 3=t\cdot\ln 1.4 \\ t=(\ln 3)/(\ln 1.4)\approx3.265 \end{gathered}`

Round your answer to the nearest tenth of an hour.

So, t = 3.3 hours