a population of insects increases at a rate of 1.5% per day. about how long will it take the population to…

Explanation:

Step1: Set up the exponential - growth formula

The formula for exponential growth is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the growth rate per period, and $t$ is the number of periods. We want to find the time $t$ when the population doubles, so $A = 2P$ and $r=0.015$. Substituting these values into the formula gives $2P=P(1 + 0.015)^t$.

Step2: Simplify the equation

Divide both sides of the equation $2P=P(1 + 0.015)^t$ by $P$ (since $P\neq0$). We get $2=(1.015)^t$.

Step3: Take the natural - logarithm of both sides

$\ln(2)=\ln(1.015^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we can rewrite the right - hand side as $t\ln(1.015)$. So, $\ln(2)=t\ln(1.015)$.

Step4: Solve for $t$

$t=\frac{\ln(2)}{\ln(1.015)}$. We know that $\ln(2)\approx0.693$ and $\ln(1.015)\approx0.0149$. Then $t=\frac{0.693}{0.0149}\approx46.6$.

Answer:

46.6 days