Suppose a certain population obeys the logistic equation dy dt = ry[1 − (y/k)]. (a) if y0 = k/3, find the time τ at which the initial population has doubled. find the value of τ corresponding to r = 0.025 per year. (b) if y0/k = α, find the time t at which y(t )/k = β, where 0 < α, β < 1. observe that t → ∞ as α → 0 or as β → 1. find the value of t for r = 0.025 per year, α = 0.1 and β = 0.9.

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

ab will be equal to