The population of foxes in a certain region over a 2-year period is estimated to be
P1(t) = 300 + 50 sin(πt/12)
in month t, and the population of rabbits in the same region in month t is given by
P2(t) = 4000 + 400 cos(πt/12)
.
Find the rate of change of the populations when t = 4. (Express a decrease in population as a negative rate of change. Round your answers to one decimal place.)
7 wabbits
To find the rate of change of the populations when t = 4, we need to differentiate the given population functions with respect to time (t) and evaluate them at t = 4.
Let's start with the population of foxes, P1(t) = 300 + 50 sin(πt/12).
To find the derivative of P1(t), we can apply the chain rule of differentiation. The derivative of sin(u) is cos(u), and since u = πt/12, the derivative of sin(πt/12) with respect to t is cos(πt/12) * π/12.
So, dP1(t)/dt = 50 * π/12 * cos(πt/12), which simplifies to (5π/6) * cos(πt/12).
Now, let's find the population of rabbits, P2(t) = 4000 + 400 cos(πt/12).
Similar to before, we can find the derivative of P2(t) using the chain rule. The derivative of cos(u) is -sin(u), and since u = πt/12, the derivative of cos(πt/12) is -sin(πt/12) * π/12.
Therefore, dP2(t)/dt = -400 * π/12 * sin(πt/12), which simplifies to (-10π/3) * sin(πt/12).
Finally, we can evaluate the derivatives at t = 4.
For the rate of change of the fox population, we substitute t = 4 into dP1(t)/dt:
dP1(4)/dt = (5π/6) * cos(π*4/12) = (5π/6) * cos(π/3) ≈ 2.89.
For the rate of change of the rabbit population, we substitute t = 4 into dP2(t)/dt:
dP2(4)/dt = (-10π/3) * sin(π*4/12) = (-10π/3) * sin(π/3) ≈ -17.21.
Therefore, the rate of change of the fox population when t = 4 is approximately 2.89, and the rate of change of the rabbit population is approximately -17.21.
Hmmmm. precalculus. Wondering if you have had basic derivatives...
if so, p1' (rate)=50PI/12 cos(piT/12) put in t=4
and p2'(rate)=-400pi/12 sinPIt/12
Now if you have to do this with limits (UGH UGH UGH).
rate=(p1(t+dt)-p(t)/dt) lim as dt>>0
rate=(300 + 50 sin(πt/12+pidt/12)-300-50sin(Pit/12) /dt
now go use the sin(A+B) formula, and reduce it...fun, fun, fun and a tablet of paper...) It will reduce to the above.