The average cost of manufacturing a quantity q of a good, is defined to be
a(q) = C(q)/q.
The average cost per item to produce q items is given by
a(q) = 0.01q2 − 0.6q + 13, for q >0.
(a) What is the total cost, C(q),of producing q goods?
For this do I just put the given equation over q and then get rid of a q from the numerator to not have a denominator?
What is the minimum marginal cost?
What is the practical interpretation of this result?
The marginal cost is at a minimum when the (blank)per item is $ (blank)
I'm assuming that the first blank is total cost and that the second blank is the answer to part a?
(c) At what production level is the average cost a minimum? What is the lowest average cost?
Compute the marginal cost at
q = 30.
I'm so sorry for all the questions. I'm really having trouble understanding this topic.
No problem at all! I'm here to help you understand.
(a) To find the total cost, C(q), you can multiply the average cost per item, a(q), by the quantity q. In this case, the given equation for a(q) is: a(q) = 0.01q^2 - 0.6q + 13.
So, the total cost, C(q), can be expressed as: C(q) = a(q) * q.
You don't need to get rid of the q in the numerator. Instead, substitute the given equation for a(q) into the expression for C(q) and simplify. Here's how to do it step by step:
C(q) = a(q) * q
= (0.01q^2 - 0.6q + 13) * q
= 0.01q^3 - 0.6q^2 + 13q
So, the total cost of producing q goods is given by C(q) = 0.01q^3 - 0.6q^2 + 13q.
Now let's move on to the next part.
(b) The minimum marginal cost is found by taking the derivative of the average cost function with respect to q and setting it equal to zero. The marginal cost is the rate of change of the cost function with respect to quantity. In this case, the average cost function is: a(q) = 0.01q^2 - 0.6q + 13.
To find the derivative, differentiate the average cost function with respect to q:
a'(q) = d(a(q))/dq
= 0.02q - 0.6
Setting this derivative equal to zero and solving for q will give you the value of q at which the marginal cost is minimized:
0.02q - 0.6 = 0
0.02q = 0.6
q = 0.6 / 0.02
q = 30
So, the minimum marginal cost is achieved at q = 30.
(c) To find the production level at which the average cost is minimized, you need to find the value of q that corresponds to the minimum point of the average cost function, a(q) = 0.01q^2 - 0.6q + 13.
Since this is a quadratic function, the minimum occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula: x = -b/2a, where a and b are the coefficients of the quadratic equation.
In this case, a = 0.01 and b = -0.6. Plug these values into the formula:
q = -(-0.6) / (2 * 0.01) = 30
So, the production level at which the average cost is minimized is q = 30.
To find the lowest average cost, substitute this value of q into the average cost function:
a(30) = 0.01(30)^2 - 0.6(30) + 13 = 9.5
Therefore, the lowest average cost is $9.50 per item.
(d) To compute the marginal cost at q = 30, you can use the derivative of the average cost function, which we found earlier:
a'(q) = 0.02q - 0.6
Plug in q = 30 into this derivative:
a'(30) = 0.02(30) - 0.6 = 0.6 - 0.6 = 0
The marginal cost at q = 30 is 0.
I hope this helps clarify the steps and answers your questions. Let me know if you need any further explanation!