Respuesta :
Answer:
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
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 = 69.2246
- Doubling period = 7.0902 minutes
- Time taken by bacteria to reach 12000 = 52.7334 minutes
- Population after 105 minutes = 1,987,397.71
The formula used to calculate the population growth:
[tex]\text{P = P}_0 (1+r)^t[/tex]
Where,
P = 300
t = 15
Now, substituting the values:
300 = P₀ x [tex](1+r)^{15}[/tex]
1300 = P₀ x[tex](1+r)^{30}[/tex]
Taking common P₀ in both the equations, we get:
[tex]\dfrac{300}{(1+r)^{15}}&=\dfrac{1300}{(1+r)^{30}}[/tex]
[tex]\dfrac {(1+r)^{30}}{(1+r)^{15}}&=\dfrac{1300}{300}[/tex]
[tex]{(1+r)^{15}}&=4.333\\\\1+r &= 1.1027\\\\r &= 0.1027[/tex]
From the above equation, the value of P₀ can be found as:
[tex]300&= \text{P}_0 \times (1+0.1027)^{15}\\\text{P}_0 &=\dfrac {300}{1.1027}^{15} &=69.2246\\[/tex]
The doubling period of culture, we have P/P₀ = 2, we have:
[tex](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[/tex]
Now, the population after 105 minutes:
[tex]\text{P} = 69.2246 \times (1+0.1027)^{105} = 1,987,397.71[/tex]
Since the population reaches 12000, we get:
[tex]\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}[/tex]
The time taken is 52.7334 minutes.
To know more about bacterial cell culture, refer to the following link:
https://brainly.com/question/1553857
