The length (in centimeters) of a typical Pacific halibut t years old is approximately
f(t) = 190(1 − 0.955e−0.25t).
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
To find the rate at which 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 and evaluate it at t=9.
First, let's find the derivative of f(t):
f'(t) = [190 * (1 - 0.955e^(-0.25t))]'
To differentiate this function, we use the chain rule. The derivative of e^(u) with respect to u is e^(u) * du/dx. In this case, u = -0.25t, so du/dt = -0.25. Thus, the derivative of e^(-0.25t) with respect to t is -0.25 * e^(-0.25t).
Now we can differentiate the function f(t) using the chain rule:
f'(t) = [190 * (1 - 0.955e^(-0.25t))]'
= 190 * [(-0.955e^(-0.25t))' + (1)' ]
= 190 * [0.955 * (-0.25) * e^(-0.25t) + 0]
= -47.875 * e^(-0.25t)
To find the rate of increase at t=9, we substitute t=9 into f'(t):
f'(9) = -47.875 * e^(-0.25*9)
To compute this value, we can use a scientific calculator or a mathematical software. The result will give you the rate of increase in centimeters per year.
Now, to find the maximum length a typical Pacific halibut can attain, we need to determine the limit of f(t) as t approaches infinity.
Taking the limit as t approaches infinity:
lim(t->∞) f(t) = lim(t->∞) 190(1 - 0.955e^(-0.25t))
As t approaches infinity, the term e^(-0.25t) goes to 0, since the exponential function decreases exponentially as t increases. Therefore, the limit is:
lim(t->∞) f(t) = 190(1 - 0.955 * 0)
= 190(1 - 0)
= 190
Therefore, the maximum length a typical Pacific halibut can attain is 190 cm.