Exponential and Logarithmic Models

Question

The count in a bacteria culture was 800 after 10 minutes and 1100 after 40 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?   

Find the doubling period.   

Find the population after 115 minutes.   

When will the population reach 14000.   

You may enter the exact value or round to 2 decimal places.

Answer 1

Let N(t) = number of bacteria after t minutes. N(t) = Ae^kt.

N(10) = Ae^10k = 800 and N(40) = Ae^40k = 1100

Since Ae^10k = 800, A = 800 / e^10k. So, (800 / e^10k)e^40k = 800e^30k = 1100.

Therefore, e^30k = 11/8.

30k = ln(11/8) So, k = 0.010615.

Ae^10(0.010615) = 800, A = initial population ≈ 719

So, N(t) ≈ 719e^0.010615t

To find the doubling time, set N(t) = 2(719) and solve for t.

N(115) = population in 115 minutes.

The population is 14000 when N(t) = 14000

Answer 2

I used exponential curve-fitting software. The initial number of bacteria is 719.

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The time it takes for the bacterial population to double is:

y = 719.4309 * 1.0107^x

2 * 719.4309 = 719.4309 * 1.0107^x

( 2 ) = 1.0107^x

x = log (2) / log (1.0107) ≈ 65.126 minutes

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y = 719.4309 * 1.0107^x

y = 719.4309 * 1.0107¹¹⁵

2,447 will be the bacterial population to the nearest whole bacteria after 115 minutes.

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y = 719.4309 * 1.0107^x

14,000 = 719.4309 * 1.0107^x

(14,000/719.4309) = 1.0107^x

x = log (14,000/719.4309) / log ( 1.0107 )

It will take about 278.898 minutes for the bacterial population to reach 14,000 according to this model.

That's about 4.65 hours.