Question

Part I On a certain university campus there is an infestation of Norway rats. It is estimated that the number of rats on campus will follow a logistic model of the form P(t)=50001+Be−kt. A) It is estimated that there were 500 rats on campus on January 1, 2010 and 750 on April 1, 2010. Using this information, find an explicit formula for P(t) where t is years since January 1, 2010. (Assume April 1, 2010 is t=.25.

Answer 1

Answer:

Step-by-step explanation:

$P(t)=50001+Be^{-kt}$

If t is years since January 1, 2010, then on January 1, 2010, t = 0.

On April 1, 2010, there are 3 months from January 1, 2010. t = 0.25 (3 months ÷ 12 months)

At t = 0, P(t) = 500

$P(0)=500 = 50001+Be^{-k\times0} = 50001+B$

$B = 500 - 50001 = -49501$

At t = 0.25, P(t) = 750

$P(0)=750 = 50001-49501e^{-k\times0.25}$

$49501e^{-0.25k} = 50001$

$e^{-0.25k} =\dfrac{50001}{49501} = 1.0101$

$-0.25k = \ln1.0101 = 0.01$

$k = -\dfrac{0.01}{0.25} = -0.04$

Substituting for B and k in P(t),

$P(t)=50001-49501e^{-0,04t}$