Question

question 2 · 1 point
a population of kangaroos is growing at a rate of 2% per year, compounded continuously. 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? round your answer up to the nearest whole number, and do not include units.
provide your answer below:

Explanation:

Step1: Recall the continuous - growth formula

The formula for continuous growth is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the growth rate, and $t$ is the time. We know that $A = 1.5P$ (since we want the population to reach 150% of its current size) and $r=0.02$.

Step2: Substitute values into the formula

Substitute $A = 1.5P$ and $r = 0.02$ into $A = Pe^{rt}$:
$1.5P=Pe^{0.02t}$. Divide both sides of the equation by $P$ (since $P
eq0$), we get $1.5 = e^{0.02t}$.

Step3: Take the natural logarithm of both sides

Take the natural - logarithm of both sides: $\ln(1.5)=\ln(e^{0.02t})$. Since $\ln(e^{x}) = x$, the right - hand side simplifies to $0.02t$. So, $\ln(1.5)=0.02t$.

Step4: Solve for $t$

We know that $\ln(1.5)\approx0.4055$. Then $t=\frac{\ln(1.5)}{0.02}$. Substitute $\ln(1.5)\approx0.4055$ into the equation: $t=\frac{0.4055}{0.02}=20.275$.

Step5: Round up

Round up $20.275$ to the nearest whole number. So, $t = 21$.

Answer:

21