Question

The population of a culture of bacteria, P(t), where t is time in days, is growing at a rate that is proportional to the population itself and the growth rate is 0.2. The initial population is 10.

Answer 1

Answer

Explanation

Using the formula for the population growth:

$P(t)=P_0\cdot(1+r)^t$

where P₀ is the initial population, r is the rate of growth, and t is the time.

From the given information, we know that:

• P₀ = 10

,

• r = 0.2

1.

And we are asked to find P(50) (when t = 50), thus, by replacing the values we get:

$P(50)=10\cdot(1+0.20)^{50}$$P(50)\approx91004.3815$

2.

For the population to double, this would mean that P(t) = 2P₀. By replacing this we get:

$2P_0=10e^{0.20t}$$2(10)=10e^{0.20t}$$20=10e^{0.20t}$$\frac{20}{10}=e^{0.20t}$$\ln\frac{2}{1}=\ln e^{0.20t}$$\ln2=0.20t$$t=\frac{\ln2}{0.20}\approx3.5days$