Calc/Finding growth rate of Bacteria

Question

A bacteria culture starts with  bacteria and grows at a rate proportional to its size. After  hours there will be  bacteria.

(a) Express the population  after  hours as a function of . Be sure to keep at least 4 significant figures on the growth rate.=   (insert answer)

(b) What will be the population after 4 hours? (insert answer)   bacteria

(c) How long will it take for the population to reach 2670? Give your answer accurate to at least 2 decimal places. (insert answer)  hours

Answer 1

Growth rate equation: P = P₀e^kt

P = population after time t

^P₀ = starting population

k = growth rate

t = time in hours

a. Find the function P(t).

In order to do this, we need to find k.

P = 720

P₀ = 360

k = ?

t = 2

P = P₀e^kt plug the values in and solve for k.

720 = 360e^k(2)

720/360 = e^2k

2 = e^2k

Use exponential ⇔ log rule: x=a^y ⇔ y =log_a(x)

x = 2, a = e, y = 2k

2k = log_e(2)

2k = ln(2)

k = ln(2)/2

k = 0.34657

Since we know the value of k now, P is now a function of t.

P(t) = 360e^0.3466t

b. What will be the population after 4 hr?

P(t) = 360e^0.3466t

P(4) = 1440.15

After 4 hours, the population will be about 1440 bacteria.

c. How long will it take for the population to reach 2670 bacteria?

We set P(t) equal to 2670 and solve for t.

P(t) = 360e^0.3466t

2670 = 360e^0.3466t

2670/360 = e^0.3466t

7.4167 = e^0.3466t

Use exponential ⇔ logarithm equation:

x = a^y ⇔ y = log_a(x)

x = 7.4167, a = e, y= 0.3466t

0.3466t = log_e(7.4167)

0.3466t = ln(7.4167)

t = ln(7.4167) / 0.3466

t = 5.782

After 5.78 hours, the bacteria reaches a population of 2670.