The reproduction function for a sardine is
f(p) = −0.0004p^2+3p, where p and f(p) are in hundred metric tons. Find the population that gives the maximum sustainable yield, and the size of the yield.
population=_________ hundred metric tons
size of yield=__________ hundred metric tons
what do you mean by "The reproduction function"?
is that the rate of change of the population, or some other weird metric?
To find the population that gives the maximum sustainable yield and the size of the yield, we need to find the maximum point of the reproduction function f(p) = -0.0004p^2 + 3p.
To do this, we can apply calculus and find the critical point by setting the derivative of the function equal to zero:
f'(p) = -0.0008p + 3 = 0
Solving this equation, we get:
-0.0008p + 3 = 0
-0.0008p = -3
p = -3 / -0.0008
p ≈ 3750
Now, we have obtained the critical point as p ≈ 3750. To determine if it is a maximum or minimum point, we can take the second derivative of the function:
f''(p) = -0.0008
Since the second derivative is negative, this indicates that the critical point is a maximum point.
Therefore, the population that gives the maximum sustainable yield is approximately 3750 hundred metric tons.
To determine the size of the yield, we substitute this population value back into the reproduction function:
f(3750) = -0.0004(3750)^2 + 3(3750)
≈ -0.0004(14,062,500) + 11,250
≈ -5,625 + 11,250
≈ 5,625
Therefore, the size of the yield at the population of approximately 3750 hundred metric tons is approximately 5625 hundred metric tons.