Mathematics, 05.05.2020 04:49 dondre54

The count in a bacteria culture was 300 after 15 minutes and 1300 after 30 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period?

Find the population after 105 minutes?

When will the population reach 12000

Answers: 2

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Answers

A bacteria culture is grown and tested to find harmful or toxic bacteria in the body or given substance. It is a method of multuplying the microbial organisms by letting them reproduce under controlled laboratory conditions.

The correct answers are:

Initial size of the culture = 69.2246Doubling period = 7.0902 minutesTime taken by bacteria to reach 12000 = 52.7334 minutesPopulation after 105 minutes = 1,987,397.71

The formula used to calculate the population growth:

$\text{P = P}_0 (1+r)^t$

Where,

P = 300

t = 15

Now, substituting the values:

300 = P₀ x $(1+r)^{15}$

1300 = P₀ x $(1+r)^{30}$

Taking common P₀ in both the equations, we get:

$\dfrac{300}{(1+r)^{15}}&=\dfrac{1300}{(1+r)^{30}}$

$\dfrac {(1+r)^{30}}{(1+r)^{15}}&=\dfrac{1300}{300}$

${(1+r)^{15}}&=4.333\\\\1+r &= 1.1027\\\\r &= 0.1027$

From the above equation, the value of P₀ can be found as:

$300&= \text{P}_0 \times (1+0.1027)^{15}\\\text{P}_0 &=\dfrac {300}{1.1027}^{15} &=69.2246\\$

The doubling period of culture, we have P/P₀ = 2, we have:

$(1+0.1027)^t = 2\\\\\text{log}(1.1027^{t}) = \text{log}(2)\\\\t\times \text{log}(1.1027) = \text{log}(2)\\\\t&= \dfrac{\text{log 2}}{\text{log 1} (1.1027)}\\\\t&=7.0902$

Now, the population after 105 minutes:

$\text{P} = 69.2246 \times (1+0.1027)^{105} = 1,987,397.71$

Since the population reaches 12000, we get:

$\begin{aligned}12000&= 69.2246 \times (1+0.1027)^t\\\\(1.1027)^t &= \dfrac{12000}{69.2246}\\\\\text{log}(1.1027^{t}) &= \text{log}(173.3488)\\\\\text t\times \text{log}(1.1027) &= \text{log}(173.3488)\\\text t &= \text{log}\dfrac{(173.3488)}{\text{log}(1.1027)}\end{aligned}$

The time taken is 52.7334 minutes.

To know more about bacterial cell culture, refer to the following link:

link

Answer from: noobieplayerxd

Inicial size of the culture = 69.2246

Doubling period = 7.0902 minutes

Population after 105 minutes = 1,987,397.71

Time for population reaches 12000: 52.7334 minutes

Step-by-step explanation:

First we need to find the exponencial function using the two information given. The model for an exponencial function is:

P = Po * (1+r)^t

Where P is the final value, Po is the inicial value, r is the rate and t is the time. So we have that:

300 = Po * (1+r)^15

1300 = Po * (1+r)^30

Isolating Po in both equations, we have that:

300/(1+r)^15 = 1300/(1+r)^30

(1+r)^30/(1+r)^15 = 1300/300

(1+r)^15 = 4.3333

1+r = 1.1027

r = 0.1027

From the first equation, we can use r to find Po:

300 = Po * (1+0.1027)^15

Po = 300 / (1.1027)^15 = 69.2246

To find the doubling period, we have that P/Po = 2, so:

(1+0.1027)^t = 2

log(1.1027^t) = log(2)

t*log(1.1027) = log(2)

t = log(2)/log(1.1027) = 7.0902 minutes

The population after 105 minutes is:

P = 69.2246 * (1+0.1027)^105 = 1,987,397.71

When the population reaches 12000:

12000 = 69.2246 * (1+0.1027)^t

(1.1027)^t = 12000/69.2246

log(1.1027^t) = log(173.3488)

t*log(1.1027) = log(173.3488)

t = log(173.3488)/log(1.1027) = 52.7334 minutes

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The count in a bacteria culture was 300 after 15 minutes and 1300 after 30 minutes. Assuming the cou...