The length (in centimeters) of a typical Pacific halibut t years old is approximately
f(t) = 210(1 − 0.94e^−0.2t).
(a) What is the length of a typical 9-year-old Pacific halibut?
cm
(b) How fast is the length of a typical 9-year-old Pacific halibut increasing?
cm/yr
(c) What is the maximum length a typical Pacific halibut can attain?
cm
a) plug in t=9
b) take the derivative
c) as t gets larger, e^-.2t gets smaller
so .94e^-.2t gets smaller
so you are left with
f(t) = 210(1 - 0) = 210
b) Take derivative and plug in 9?
To find the answers to these questions, we will use the given function f(t) = 210(1 − 0.94e^−0.2t), where t represents the age of the Pacific halibut.
(a) To find the length of a typical 9-year-old Pacific halibut, substitute t = 9 into the function f(t) and solve for f(9):
f(9) = 210(1 − 0.94e^−0.2(9))
First, calculate e^−0.2(9):
e^−0.2(9) ≈ 0.41111
Next, substitute this value back into the equation:
f(9) = 210(1 − 0.94 * 0.41111)
f(9) ≈ 210(1 − 0.38535)
f(9) ≈ 210 * 0.61465
f(9) ≈ 128.877
Therefore, the length of a typical 9-year-old Pacific halibut is approximately 128.877 cm.
(b) To find how fast the length of a typical 9-year-old Pacific halibut is increasing, we need to find the derivative of the function f(t) with respect to t. This will give us the rate of change or the instantaneous rate of change of the function at that specific point.
f(t) = 210(1 − 0.94e^−0.2t)
To differentiate f(t), use the product rule and chain rule:
f'(t) = 210 * (-0.94)(-0.2)e^(-0.2t)
Substituting t = 9 into f'(t):
f'(9) = 210 * (-0.94)(-0.2)e^(-0.2(9))
Calculate e^(-0.2(9)) as before:
e^(-0.2(9)) ≈ 0.41111
Therefore:
f'(9) = 210 * (-0.94)(-0.2) * 0.41111
f'(9) ≈ 18.334
The length of a typical 9-year-old Pacific halibut is increasing at a rate of approximately 18.334 cm/yr.
(c) To find the maximum length a typical Pacific halibut can attain, we need to find the limit of the function f(t) as t approaches infinity.
lim(t→∞) f(t) = lim(t→∞) 210(1 − 0.94e^−0.2t)
As t approaches infinity, e^−0.2t approaches 0. Therefore, the limit of the function is:
lim(t→∞) f(t) = 210(1 - 0.94 * 0)
lim(t→∞) f(t) = 210
Therefore, the maximum length a typical Pacific halibut can attain is 210 cm.