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