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

Explanation:

Step1: Use the compound - growth formula

The compound - growth formula 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 (t) when (A = 2P) and (r=0.015). Substituting (A = 2P) into the formula gives (2P=P(1 + 0.015)^t). Divide both sides by (P) (since (P\neq0)), we get (2=(1.015)^t).

Step2: Take the natural logarithm of both sides

Taking the natural logarithm of both sides: (\ln(2)=\ln((1.015)^t)). Using the property of logarithms (\ln(a^b)=b\ln(a)), we have (\ln(2)=t\ln(1.015)).

Step3: Solve for (t)

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

Answer:

46.6 days