Bacterial Growth A colony of bacteria grows according to the law of uninhibited growth

Question

Bacterial Growth

A colony of bacteria grows according to the law of uninhibited growth according to the function , where N(t) = [latex]100e^{0.045(t)}[/latex] is measured in grams and t is measured in days.
(a) Determine the initial amount of bacteria.
(b) What is the growth rate of the bacteria?
(c) What is the population after 5 days?
(d) How long will it take for the population to reach 140 grams?
(e) What is the doubling time for the population?

Accepted Answer

(a) The intial amount of bacteria, [latex] N_{0}[/latex], is obtained when t = 0, so

[latex]N_{0}=N(0){=}100e^{0.045(0)}=100\,\mathrm{grams}[/latex]

(b) Compare N(t) = [latex]100e^{0.045(t)}[/latex] to N(t) [latex]= N_{0}e^{kt}.[/latex] The value of k, 0.045, indicates a growth rate of 4.5%.

(c) The population after 5 days is N(5) = [latex]100e^{0.045(5)} \approx 125.2 [/latex] grams.

(d) To find how long it takes for the population to reach 140 grams, solve the equation N(t) = 140

[latex]100e^{0.045(t)} = 140[/latex]

[latex]e^{0.045(5)} = 1.4[/latex] Divide both sides of the equation by 100.

[latex] 0.045t = \ln 1.4 [/latex] Rewrite as a logarithm.

[latex] t = \frac{\ln 1.4}{0.045}[/latex] Divide both sides of the equation by 0.045.

≈ 7.5 days

The population reaches 140 grams in about 7.5 days.

(e) The population doubles when N(t) = 200 gram, so the doubling time can be found by solving the equation 200 = [latex]100e^{0.045t} [/latex] for t.

200 = [latex]100e^{0.045t}[/latex]

2 = [latex]e^{0.045t} [/latex] Divide both sides of the equation by 100.

[latex] \ln 2 = 0.045t [/latex] Rewrite as a logarithm

[latex] t = \frac{\ln 2 }{0.045}[/latex] Divide both sides of the equation by 0.045

≈ 15.4 days.

The population doubles approximately every 15.4 days.

Question 4.4.8.1: Bacterial Growth A colony of bacteria grows according to the......

This Question has been Answered.

Already have an account?

Login

Related Answered Questions

Wood products The EFISCEN wood product model classifies wood products according to their ...

Verified Answer:

fruit fly population Fruit flies are placed in a half-pint milk bottle with a banana ...

Verified Answer:

Using newton’s law of Cooling An object is heated to 100°C (degrees Celsius) and is then ...

Verified Answer:

estimating the age of ancient Tools Traces of burned wood along with ancient stone tools ...

Verified Answer:

Bacterial Growth A colony of bacteria increases according to the law of uninhibited ...

Verified Answer:

Writing expressions as a Single logarithm Write each of the following as a single ...

Verified Answer:

Fitting a logistic function to Data The data in Table 11 represent the amount of yeast ...

Verified Answer:

Writing a logarithmic Expression as a Sum and Difference of Logarithms Write loga √x²+1/x³(x+1)^4’ x>0 as a sum and difference of logarithms. Express all powers as factors. ...

Verified Answer:

Approximating a logarithm Whose Base is neither 10 Nor e Approximate log2 7. Round the answer to four decimal places. ...

Verified Answer:

Using the Change-of-Base formula Approximate: (a) log5 89 (b) log√2 √5 Round answers to four decimal places. ...

Verified Answer: