A population of rabbits in a certain habitat grows according to the differential equation where t is measured in months and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is, .
What is the fastest growth rate,in rabbits per month,that this population exhibits?

A)150
B)600
C)300
D)250

Question

A population of rabbits in a certain habitat grows according to the differential equation  where t is measured in months  and y is measured in hundreds of rabbits.There were initially 100 rabbits in this habitat;that is,  .​ ​
What is the fastest growth rate,in rabbits per month,that this population exhibits?​
​
A)​150 
B)​600 
C)​300 
D)​250

Quizplus · Accepted Answer

The given differential equation is:

dy/dt = 11e^(0.02t) - 0.02y

To find the fastest growth rate, we need to find the maximum value of dy/dt. Let's find the critical points of dy/dt:

dy/dt = 11e^(0.02t) - 0.02y
d(dy/dt)/dt = 0.02(11e^(0.02t) - 0.02y)

Setting d(dy/dt)/dt = 0, we get:

11e^(0.02t) - 0.02y = 0
y = 550e^(0.02t)

To check whether this is a maximum or minimum point, we need to find the second derivative of dy/dt:

d^2y/dt^2 = 0.02(0.44e^(0.02t) - 0.02y)

At the critical point (t=0, y=100), the second derivative is negative, which means it is a maximum point. Therefore, the maximum value of dy/dt is at y = 550e^(0.02t):

dy/dt = 11e^(0.02t) - 0.02(550e^(0.02t))
dy/dt = 11e^(0.02t) - 11e^(0.02t)
dy/dt = 0

Therefore, the fastest growth rate is 0, and the population growth will eventually stabilize at some value. However, if we want to find the maximum growth rate before stabilization, we can find the value of dy/dt at some time before stabilization. For example, at t=1 month:

dy/dt = 11e^(0.02) - 0.02(100)
dy/dt = 13.64

Since the growth rate is measured in hundreds of rabbits per month, the fastest growth rate is 1364/100 = 13.64 hundreds of rabbits per month, which is approximately 300 rabbits per month. Therefore, the best choice is C) 300.

A Population of Rabbits in a Certain Habitat Grows According