Question

The number of bacteria in a culture is given by the function n(t)=990e^0.45t where t is measured in hours (a) What is the exponential rate of growth of this bacterium population?Your answer is % (b) What is the initial population of the culture (at t=0)?Your answer is (c) How many bacteria will the culture contain at time t=7?Your answer is

Answer 1

.

Answer:

The function is given below as

`n(t)=990e^(0.45t)`

The exponential function is given below as

`\begin{gathered} A(t)=A_0e^(rt) \\ t=time \\ A_0=initial\text{ number } \\ A(t)=number\text{ after time t} \\ r=rate \end{gathered}`

By comparing coefficients, we will have

`\begin{gathered} A_0=990 \\ r=0.45 \end{gathered}`

Hence,

the rate will be

`r=0.45*100=45\%`

Therefore,

The rate is

`\Rightarrow45\%`

Step 2:

To figure out the initial population, we will substitute t=0 in the function above

`\begin{gathered} n(t)=990e^(0.45t) \\ n(0)=990e^(0.45*0) \\ n(0)=990e^0 \\ n(0)=990 \end{gathered}`

Hence,

The initial population of the culture is

`\Rightarrow990`

Step 3:

To figure out the number of bacteria at t=7, we will have

`\begin{gathered} n(t)=990e^(0.45t) \\ n(7)=990e^(0.45*7) \\ n(7)=990e^(3.15) \\ n(7)=990*23.3361 \\ n(7)=23102.7 \\ n(7)=23103 \end{gathered}`

Hence,

The number of bacteria at time t=7 will be

`\Rightarrow23103`