Question

the function ( p(t) = \frac{2000t}{4t + 75} ) gives the population ( p ) of deer in an area after ( t ) months.

a) find ( p(9) ), ( p(45) ), and ( p(90) )
b) find ( p(9) ), ( p(45) ), and ( p(90) )
c) interpret the meaning of your answers to part (a) and (b) what is happening to the population of deer in the long term?

a) ( p(9) = 12.174 ) deer/month
(type an integer or decimal rounded to three decimal places as needed )

( p(45) = square )
(type an integer or decimal rounded to three decimal places as needed )

Explanation:

Step1: Find the derivative of \( p(t) \)

We use the quotient rule: if \( p(t)=\frac{u(t)}{v(t)} \), then \( p'(t)=\frac{u'(t)v(t)-u(t)v'(t)}{v(t)^2} \). Here, \( u(t) = 2000t \), so \( u'(t)=2000 \); \( v(t)=4t + 75 \), so \( v'(t)=4 \).

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Step2: Calculate \( p'(45) \)

Substitute \( t = 45 \) into \( p'(t) \):

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Answer:

\( p'(45)\approx2.307 \)