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) raised to 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?
you want m where
19.3(1.089^m) = 138
1.089^m = 7.150
m log1.089 = log7.150
m = log 1.089 / log 7.150
what's the answer
To find out how many months have passed since the plants were first introduced, we need to solve the equation p(m) = 138 for m.
The given equation for the number of plants after m months is p(m) = 19.3(1.089)^m.
Substituting 138 for p(m), we get:
138 = 19.3(1.089)^m
To solve this equation for m, we need to isolate the variable m.
First, divide both sides of the equation by 19.3:
138/19.3 = (1.089)^m
Next, take the natural logarithm (ln) of both sides of the equation to eliminate the exponent:
ln(138/19.3) = ln((1.089)^m)
Using the property of logarithms, we can bring down the exponent m:
ln(138/19.3) = m * ln(1.089)
Finally, divide both sides of the equation by ln(1.089) to solve for m:
m = ln(138/19.3) / ln(1.089)
Using a calculator, we can compute this value:
m ≈ 6.407
Therefore, approximately 6.407 months have passed since the plants were first introduced.