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: Write the continuous - growth formula

The formula for continuous exponential 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 the population reaches 150% of its current size) and $r=0.02$. Substituting these values into the formula gives $1.5P = Pe^{0.02t}$.

Step2: Simplify the equation

Divide both sides of the equation $1.5P = Pe^{0.02t}$ by $P$ (since $P
eq0$). We get $1.5=e^{0.02t}$.

Step3: Take the natural - logarithm of both sides

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

Step4: Solve for $t$

We can solve for $t$ by dividing both sides of the equation $\ln(1.5)=0.02t$ by $0.02$. So, $t=\frac{\ln(1.5)}{0.02}$.

Step5: Calculate the value of $t$

We know that $\ln(1.5)\approx0.4055$ and $\frac{\ln(1.5)}{0.02}=\frac{0.4055}{0.02}=20.275$. Rounding up to the nearest whole number, $t = 21$.

Answer:

21