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.
O<Q<2/3 AND 2/3<Q
wrt
4Q^2+12Q+50
How can solve
32q+10
(i) To determine the values of q for which ATC (Average Total Cost) is falling, we need to find where the derivative of ATC is negative.
Let me calculate it for you:
The ATC is given by the equation ATC = C / q, where C is the total cost function.
Differentiating the ATC equation with respect to q:
d(ATC)/dq = (dC/dq * q - C) / q^2
Substituting the given total cost function, C = 16q^2 + 10q + 36:
d(ATC)/dq = ((32q + 10) * q - (16q^2 + 10q + 36)) / q^2
Now, let's determine the values of q for which d(ATC)/dq is negative:
((32q + 10) * q - (16q^2 + 10q + 36)) / q^2 < 0
Simplifying and rearranging the inequality, we get:
16q^2 - 614q - 36 > 0
To find the values of q for which ATC is falling, we need to solve this quadratic inequality.
(ii) Similarly, to determine the values of q for which ATC is rising, we just need to find where the derivative of ATC is positive.
Let's solve these equations to find the specific values of q for both cases.
To find the values of q for which ATC (Average Total Cost) is falling or rising, we need to analyze the relationship between ATC and output (q) based on the given total cost function C = 16q^2 + 10q + 36.
To determine whether ATC is falling or rising, we need to examine the slope of the ATC curve. If the slope of the ATC curve is negative, it means ATC is falling, and if the slope is positive, it means ATC is rising.
The ATC can be calculated by dividing the total cost (C) by the output (q):
ATC = C/q
To find the values of q for which ATC is falling, we need to find the range where the slope of the ATC curve is negative. This can be done by examining the first derivative of the ATC function. Let's differentiate the ATC function with respect to q:
d(ATC)/dq = (dC/dq - C)/(q^2)
Now, let's find the critical points where d(ATC)/dq = 0:
0 = (dC/dq - C)/(q^2)
Now, let's differentiate the total cost function C = 16q^2 + 10q + 36 with respect to q:
dC/dq = 32q + 10
Substituting the value of dC/dq into the equation:
0 = (32q + 10 - (16q^2 + 10q + 36))/(q^2)
Simplifying the equation:
0 = 32q + 10 - 16q^2 - 10q - 36
0 = -16q^2 + 22q - 26
Now, we have a quadratic equation. To find the values of q for which ATC is falling, we need to find the roots of this equation. We can solve this equation using the quadratic formula:
q = (-b ± √(b^2 - 4ac))/(2a)
In our case:
a = -16
b = 22
c = -26
Substituting these values into the quadratic formula:
q = (-22 ± √(22^2 - 4(-16)(-26)))/(2(-16))
Solving this equation will give us the values of q for which ATC is falling.
To find the values of q for which ATC is rising, we need to examine the range of values of q where the slope of the ATC curve is positive. This can be done by finding the range of q values outside the range where ATC is falling.
Once we solve the quadratic equation and find the values of q, we can determine whether ATC is falling or rising by evaluating the slope of the ATC curve at those points. If the slope is negative, ATC is falling, and if the slope is positive, ATC is rising.