Given the total cost function: C=16q2 + 10q+36 (where q is the output)
Find: (i) values of q for which ATC is falling, and
(ii) values of q for which ATC is rising
Value of q which ATC is falling and Value of which ATC is rising
economics
To find the values of q for which the average total cost (ATC) is falling or rising, we need to analyze how the ATC changes with respect to the quantity output (q).
The average total cost (ATC) is calculated by dividing the total cost (C) by the quantity output (q). So, we can express ATC as a function of q by dividing the total cost equation by q:
ATC = C/q
Given the total cost function: C = 16q^2 + 10q + 36, we substitute this into the ATC equation:
ATC = (16q^2 + 10q + 36) / q
Simplifying the ATC equation, we have:
ATC = 16q + 10 + 36/q
To determine whether ATC is falling or rising, we analyze the changes in the ATC equation with respect to q.
(i) Values of q for which ATC is falling:
ATC will be falling if its derivative (d(ATC)/dq) is negative.
To find the derivative of ATC, we differentiate the ATC equation with respect to q:
d(ATC)/dq = 16 - 36/q^2
We set the derivative equal to zero and solve for q:
16 - 36/q^2 = 0
16 = 36/q^2
q^2 = 36/16
q^2 = 9/4
q = ±√(9/4)
q = ±(3/2)
So, the values of q for which ATC is falling are q = -3/2 and q = 3/2.
(ii) Values of q for which ATC is rising:
ATC will be rising if its derivative (d(ATC)/dq) is positive.
By evaluating the sign of the derivative on each side of the values q = -3/2 and q = 3/2, we can determine the values of q for which ATC is rising.
Let's evaluate d(ATC)/dq at q = -3/2:
d(ATC)/dq = 16 - 36/((-3/2)^2) = 16 - 36/(9/4) = 16 - 16/9 = 16 - 16/9 = 128/9 - 16/9 = 112/9 > 0
Since d(ATC)/dq is positive at q = -3/2, ATC is rising at this value.
Now, let's evaluate d(ATC)/dq at q = 3/2:
d(ATC)/dq = 16 - 36/((3/2)^2) = 16 - 36/(9/4) = 16 - 16/9 = 128/9 - 16/9 = 112/9 > 0
Again, d(ATC)/dq is positive at q = 3/2, indicating that ATC is rising at this value as well.
Therefore, the values of q for which ATC is rising are q = -3/2 and q = 3/2.
Given the total cost function: C=16q2 + 10q+36 (where q is the output)
Find: (i) values of q for which ATC is falling, and
(ii) values of q for which ATC is rising