Respuesta :
Answer:
The correct answer is 20 years.
Explanation:
It is given that the growing rate is 2% or 2/100 = 0.02
This rate got increased to 150%
The formula for exponential growth is:
A = Pe^rt
Let the initial population be 100%
150 = 100.e^0.02*t
3/2 = e^0.02t
1.5 = e^0.02t
After taking log from both the sides:
ln(1.5) = 0.02t * ln(e) [ln(e) = 1]
ln(1.5) = 0.02t
t = ln(1.5)/0.02
t = 20.27
Thus, it will take around 20 years for the size of the population to reach 150 percent of its present size on the basis of the exponential growth function.
Answer:
It will take 20 years for the size of the population to reach 150% of its current size according to the exponential growth function.
Explanation:
The exponential growth function is given by:
[tex]P(t) = P(0)e^{rt}[/tex]
In which [tex]P(t)[/tex] is the population after t years, [tex]P(0)[/tex] is the initial population, e is the Euler number and r is the growth rate(decimal).
In this problem, we have that:
A population of kangaroos is growing at a rate of 2% per year, compounded continuously. This means that [tex]r = 0.02[/tex].
If the growth rate continues, how many years will it take for the size of the population to reach 150% of its current size according to the exponential growth function?
This is the value of t when [tex]P(t) = 1.5P(0)[/tex]. So
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]1.5P(0) = P(0)e^{0.02t}[/tex]
[tex]e^{0.02t} = 1.5[/tex]
Since ln and e are inverse operation, we apply e to both sides of the equation to find t.
[tex]\ln{e^{0.02t} }= \ln{1.5}[/tex]
[tex]0.02t = 0.4055[/tex]
[tex]t = \frac{0.4055}{0.02}[/tex]
[tex]t = 20.27[/tex]
Rouding to the nearest whole number, it is 20 years.
It will take 20 years for the size of the population to reach 150% of its current size according to the exponential growth function.
