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. f(x) = 500(1.07)^x (simplify your answer. use integers or decimals for any numbers in the expression.) how many years will it take for the plant to become invasive? it will take about \\(\square\\) years for the plant to become invasive. (round up to the nearest year.)

Explanation:

Step1: Set up the equation

We know the exponential growth function is \( f(x) = 500(1.07)^x \), and we want to find \( x \) when \( f(x)=10000 \). So we set up the equation:
\( 500(1.07)^x = 10000 \)

Step2: Divide both sides by 500

Divide each side of the equation by 500 to simplify:
\( \frac{500(1.07)^x}{500}=\frac{10000}{500} \)
\( (1.07)^x = 20 \)

Step3: Take the natural logarithm of both sides

Take the natural log of both sides to use the logarithm property \( \ln(a^b)=b\ln(a) \):
\( \ln((1.07)^x)=\ln(20) \)
\( x\ln(1.07)=\ln(20) \)

Step4: Solve for x

Divide both sides by \( \ln(1.07) \) to solve for \( x \):
\( x = \frac{\ln(20)}{\ln(1.07)} \)

Step5: Calculate the value

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

Step6: Round up to the nearest year

Since we need to round up to the nearest year, \( x\approx45 \)

Answer:

45