Problema Solution
The observed bunny rabbit population on am island is givin by the function p= -0.4t^2+ 130t+1200, where t is the time in months since they began observing the rabbits. When is the maximum population being attained? What is the maximum population? When does the bunny rabbit population disappear from the island?
Answer provided by our tutors
Lets observe the function
p(t) = -0.4t^2+ 130t+1200
it is a quadratic function and it will help if you draw the parabola
- maximum
Since the value in front of t^2 is negative (a = -0.4 < 0) this means that the parabola opens downwards and the vertex is the highest point the graph can get (maximum):
maximum population = c - b^2/4a = 1200 - 130^2/4*(-0.4) = 11762.5 rabits
this value is attained for t : the solution of the quadratic equation -0.4t^2+ 130t+1200 = 11762.5
t = 162.5 months
The bunny rabbit population will disappear when p(t) = 0 that is
-0.4t^2+ 130t+1200 = 0
we are considering the positive roots of the quadratic equation that is
for t1 = 333.983 months the population will disappear p(t1) = 0.
The maximum population is attained in 162.5 months.
Maximum population is 11762.5 rabbits.
The bunny rabbit population will disappear in 333.983 months.