Expert: Bobby Soltani - 10/15/2004

Question
The book gives an answer but does not explain step-by-step how the answer was attained. Can you please expain the steps for the given problem:  The rabbit population on asmall island is given by the function P(t)=130t-0.35t^4+1000.  Where t is the time in months since observation of the rabbit population began.  What is the maximum population?  

Thanks.  

Answer
Hi Jackie,

To find a maximum for this equation, you take the derivative of P(t) and set it equal to zero(Since the derivative of an equation represents the slope of the equation, it will be zero at the maximum).

P' = 130 - 1.4t^3
set it to zero

0 = 130 - 1.4t^3
add 1.4t^3 to both sides

1.4t^3 = 130
divide both sides by 1.4

t^3 = 92.86
take the cube root of both sides

t = 4.528

Therefore the maximum population occurs after about 4 and a half months.

Note that the derivative of x^n is equal to nx^(n-1).  For example
f = 3x^4
f = 3*(x^4)
f' = 3*(4x^3)
f' = 12x^3

Let me know if you have any questions.  
Bobby