Answer:
(a) 9057
(b) 28,895
(c) 14.856
(d) 240,000
Step-by-step explanation:
(a) Evaluate the expression for t=0.
p(0) = 240/(1 +25.5·1) = 240/26.5 = 9.0566
This number is in thousands, so to the nearest whole number, the initial population is 9057.
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(b) Evaluate the expression for t=5.
p(5) = 240/(1 +25.5e^-1.25) ≈ 240/8.305872 ≈ 28.8952
Again, this is thousands, so the nearest whole number is 28,895 trout in 5 years.
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(c) For this, we want to solve for t when p(t) = 148.
148 = 240/(1 +25.5e^(-.25t))
25.5e^(-.25t) = 240/148 -1 = 23/37 . . . . . . rearrange
e^(-.25t) ≈ 0.024377318 . . . . . . . . . . . . . . . divide by 25.5
-.25t ≈ ln(0.024377318) ≈ -3.714102 . . . . . .take the natural log
t ≈ 3.714102/0.25 ≈ 14.85641 . . . . . . . . . . . . divide by the coefficient of t
It will take about 14.856 years for the population to reach 148,000.
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(d) As t goes toward infinity, the exponential term goes to zero, so the fraction becomes 240/1. The long-term population is expected to be 240,000.
