Question

a plant will become invasive when the number of plants reaches 10,000. model the situation with an exponential growth function. how many years will it take for the plant to become invasive? explain how you found the solution. write the exponential growth function for the model. n(t) = \square (simplify your answer. use integers or decimals for any numbers in the expression.) initial population 500 plants annual growth rate 7%

Explanation:

Step1: Recall Exponential Growth Formula

The general formula for exponential growth is \( N(t) = N_0(1 + r)^t \), where \( N_0 \) is the initial population, \( r \) is the annual growth rate (in decimal), and \( t \) is the time in years. Here, \( N_0 = 500 \) and \( r = 0.07 \) (since 7% = 0.07). So the function is \( N(t)=500(1 + 0.07)^t=500(1.07)^t \).

Step2: Set Up Equation for Invasive Population

We want to find \( t \) when \( N(t)=10000 \). So we set up the equation \( 10000 = 500(1.07)^t \).

Step3: Solve for \( t \)

First, divide both sides by 500: \( \frac{10000}{500}=(1.07)^t \), which simplifies to \( 20=(1.07)^t \). Then, take the natural logarithm of both sides: \( \ln(20)=\ln((1.07)^t) \). Using the logarithm power rule \( \ln(a^b)=b\ln(a) \), we get \( \ln(20)=t\ln(1.07) \). Then, solve for \( t \): \( t = \frac{\ln(20)}{\ln(1.07)} \).

Step4: Calculate the Value of \( t \)

Using a calculator, \( \ln(20)\approx2.9957 \) and \( \ln(1.07)\approx0.0677 \). So \( t=\frac{2.9957}{0.0677}\approx44.25 \).

Answer:

The exponential growth function is \( N(t) = 500(1.07)^t \), and it will take approximately 44.25 years for the plant to become invasive.