Question

a type of plant is introduced into an ecosystem and quickly begins to take over. a scientist counts the number of plants after m months and develops the equation ( p(m)=19.3(1.089)^m ) to model the situation. most recently, the scientist counted 138 plants. assuming there are no limiting factors to the growth of the plants, about how many months have passed since the plants were first introduced? 6.1 6.6 7.2 23.1

Explanation:

Step1: Set up the equation

We are given the model \(P(m)=19.3(1.089)^{m}\), and we know that \(P(m) = 138\). So we set up the equation \(138=19.3(1.089)^{m}\).

Step2: Solve for \((1.089)^{m}\)

Divide both sides of the equation by \(19.3\): \(\frac{138}{19.3}=(1.089)^{m}\). Calculate \(\frac{138}{19.3}\approx7.15\). So \(7.15=(1.089)^{m}\).

Step3: Take the natural logarithm of both sides

Using the property \(\ln(a^{b})=b\ln(a)\), if \(y = a^{x}\), then \(x=\frac{\ln(y)}{\ln(a)}\). Here \(a = 1.089\) and \(y=7.15\). So \(m=\frac{\ln(7.15)}{\ln(1.089)}\).

Step4: Calculate the logarithms

We know that \(\ln(7.15)\approx1.967\) and \(\ln(1.089)\approx0.085\). Then \(m=\frac{1.967}{0.085}\approx23.1\).

Answer:

\(23.1\)