92.6k views
1 vote
A bacteria culture grows with a constant relative growth rate. After 2 hours there are 400 bacteria and after 8 hours the count is 50,000.

(a) Find the initial population. P(0) = 80 )ãbacteria
(b) Find an expression for the population after t hours. r(t) = 180( 125(6 Plt) =180(125(2))-
(c) Find the number of cells after 7 hours. (Round your answer to the nearest integer.) P(7)=72.358- bacteria
(d) Find the rate of growth after 7 hours. (Round your answer to the nearest integer.) P(7) 2x bacteria/hour
(e) When will the population reach 200,000? (Round your answer to one decimal place.) hours

User Delmania
by
7.7k points

1 Answer

1 vote

Answer:

a) P(0) = 80

b)
P(t) = 80(2.2361)^t

c) 22,363 cells.

d) The rate of growth after 7 hours is of 18,000 bacteria per hour.

e) 9.7 hours.

Explanation:

A bacteria culture grows with a constant relative growth rate.

This means that the population is given by:


P(t) = P(0)(1+r)^t

In which P(0) is the initial population and r is the growth rate, as a decimal.

After 2 hours there are 400 bacteria and after 8 hours the count is 50,000.

This means that in 6 hours, the population went from 400 bacteria to 50,000 bacteria. We use this to find r. So


50000 = 400(1+r)^6


(1+r)^6 = (50000)/(400)


(1+r)^6 = 125


\sqrt[6]{(1+r)^6} = \sqrt[6]{125}


1 + r = 125^{(1)/(6)}


1 + r = 2.2361

So


P(t) = P(0)(2.2361)^t

(a) Find the initial population. P(0)

We have that P(2) = 400. We use this to find P(0). So


P(t) = P(0)(2.2361)^t


400 = P(0)(2.2361)^2


P(0) = (400)/((2.2361)^2)


P(0) = 80

So


P(t) = 80(2.2361)^t

(b) Find an expression for the population after t hours.


P(t) = 80(2.2361)^t

(c) Find the number of cells after 7 hours.

This is P(7). So


P(7) = 80(2.2361)^7 = 22363

22,363 cells.

(d) Find the rate of growth after 7 hours.

We have to find the derivative when t = 7. So


P(t) = 80(2.2361)^t


P^(\prime)(t) = 80ln(2.2361)(2.2361)^t


P^(\prime)(7) = 80ln(2.2361)(2.2361)^7 = 18000

The rate of growth after 7 hours is of 18,000 bacteria per hour.

(e) When will the population reach 200,000?

This is t for which
P(t) = 200000. So


P(t) = 80(2.2361)^t


200000 = 80(2.2361)^t


(2.2361)^t = (200000)/(80)


(2.2361)^t = 2500


\log{(2.2361)^t} = \log{2500}


t\log{2.2361} = \log{2500}


t = \frac{\log{2500}}{\log{2.2361}}


t = 9.7

So 9.7 hours.

User Carl Poole
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories