If TC=.04q3-.9q2+10q+100. Find:
1)AC
2)MC
3)Slope of AC&MC
4)Value of q at which Average Variable Cost is minimum.
To find the answers to these questions, you need to differentiate the total cost (TC) equation with respect to q. Differentiation will help us find the average cost (AC), marginal cost (MC), and the slope of AC and MC.
Before calculating the derivatives, let's define the given equation:
TC = 0.04q^3 - 0.9q^2 + 10q + 100
1) Average Cost (AC):
AC is calculated by dividing the total cost (TC) by the quantity (q). Therefore, AC can be found by dividing the TC equation by q:
AC = (0.04q^3 - 0.9q^2 + 10q + 100) / q
= 0.04q^2 - 0.9q + 10 + 100/q
2) Marginal Cost (MC):
MC is obtained by differentiating the total cost (TC) equation with respect to q. Differentiating the equation yields the derivative of TC with respect to q, which represents the MC:
MC = d(TC)/dq
To calculate MC, differentiate each term of the TC equation separately and then combine them:
d(0.04q^3)/dq = 0.12q^2
d(-0.9q^2)/dq = -1.8q
d(10q)/dq = 10
d(100)/dq = 0
Therefore, MC = 0.12q^2 - 1.8q + 10
3) Slope of AC and MC:
The slope of AC and MC can be found by taking the derivative of AC and MC, respectively:
Slope of AC = d(AC)/dq
Slope of MC = d(MC)/dq
Differentiating AC and MC functions, we get:
d(AC)/dq = 0.08q - 0.9 - 100/q^2
d(MC)/dq = 0.24q - 1.8
Therefore, the slope of AC is 0.08q - 0.9 - 100/q^2, and the slope of MC is 0.24q - 1.8.
4) Value of q at which Average Variable Cost is minimum:
To find the value of q that minimizes the average variable cost (AVC), we need to find the value of q at which the derivative of AVC with respect to q equals zero.
AVC is given by:
AVC = AC - FC
where FC is the fixed cost.
Since we are looking for the minimum value of AVC, we need to find the critical points of AVC by setting its derivative equal to zero:
d(AVC)/dq = d(AC-FC)/dq = 0
By substituting the derivative of AC obtained earlier into the equation, we have:
0.08q - 0.9 - 100/q^2 - FC' = 0
Simplifying and solving for q will give you the value at which AVC is minimum.