Question

a logistic model is given by the equation ( p(t)=\frac{80}{1 + 3e^{-0.3t}} ). to the nearest hundredth, for what value of ( t ) does ( p(t)=40 )?

Explanation:

Step1: Substitute \(P(t) = 40\) into the equation

$$ 40=\frac{80}{1 + 3e^{-0.3t}} $$

Step2: Cross - multiply

$$ 40(1 + 3e^{-0.3t})=80 $$

$$ 1+3e^{-0.3t}=\frac{80}{40}=2 $$

Step3: Solve for \(e^{-0.3t}\)

$$ 3e^{-0.3t}=2 - 1=1 $$

$$ e^{-0.3t}=\frac{1}{3} $$

Step4: Take the natural logarithm of both sides

$$ \ln(e^{-0.3t})=\ln(\frac{1}{3}) $$

Using the property \(\ln(a^b)=b\ln(a)\), we get \(-0.3t=\ln(1)-\ln(3)\)
Since \(\ln(1) = 0\), then \(-0.3t=-\ln(3)\)

Step5: Solve for \(t\)

$$ t=\frac{\ln(3)}{0.3} $$

$$ \ln(3)\approx1.0986 $$

$$ t=\frac{1.0986}{0.3}\approx3.66 $$

Answer:

\(t = 3.66\)