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 $p(m)=19.3(1.08)^m$ 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 23.1 6.6 7.2

Explanation:

Step1: Set up the equation

We know that $P(m)=19.3(1.08)^m$ and $P(m) = 138$. So, $19.3(1.08)^m=138$.

Step2: Isolate the exponential term

Divide both sides by 19.3: $(1.08)^m=\frac{138}{19.3}\approx7.15$.

Step3: Take the natural - logarithm of both sides

$\ln(1.08^m)=\ln(7.15)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $m\ln(1.08)=\ln(7.15)$.

Step4: Solve for m

$m = \frac{\ln(7.15)}{\ln(1.08)}$. Since $\ln(7.15)\approx1.967$ and $\ln(1.08)\approx0.077$, then $m=\frac{1.967}{0.077}\approx25.545$. But if we use common - logarithms (base 10), $\log(7.15)\approx0.854$ and $\log(1.08)\approx0.033$, and $m=\frac{0.854}{0.033}\approx25.879$. A more accurate calculation with natural logarithms: $m=\frac{\ln(7.15)}{\ln(1.08)}=\frac{1.9671}{0.07696}\approx25.56$. Among the given options, the closest value is 23.1.

Answer:

23.1