The rabbit population on a small island is observed to be given by the function
P(t)=120t−0.4t4+700
where t is the time (in months) since observations of the island began.
in the words of Larry King, "What's the question?"
I bet we want the maximum, but usually the foxes are included in these island problems.
dP/dt = 0 at max = 120 -1.6 t^3
t^3 = 120
t = 4.93 months
I guess the foxes were factored in when calculating the surviving rabbit population, eh?
To find the average rate of change of the rabbit population over a given time interval, we can use the formula:
Average Rate of Change = (P(t2) - P(t1)) / (t2 - t1)
where P(t1) is the initial population at time t1 and P(t2) is the final population at time t2.
Let's say we want to find the average rate of change of the rabbit population over a time interval from t1 to t2.
1. Substitute the values of t1 and t2 into the function P(t) to find the initial and final populations:
P(t1) = 120t1 - 0.4t1^4 + 700
P(t2) = 120t2 - 0.4t2^4 + 700
2. Subtract P(t1) from P(t2) to find the difference in population:
P(t2) - P(t1) = (120t2 - 0.4t2^4 + 700) - (120t1 - 0.4t1^4 + 700)
3. Simplify the expression by combining like terms.
4. Subtract t2 from t1 to find the difference in time: t2 - t1.
5. Divide the difference in population by the difference in time to find the average rate of change:
Average Rate of Change = (P(t2) - P(t1)) / (t2 - t1)
Plug in the values of t1 and t2 to get the specific average rate of change over the given time interval.