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.
I know that the total cost is 0.01q^3-0.6q^2+13q
What is the minimum marginal cost?
Here I tried to find the derivative of total cost to get marginal cost and I got .03q^2-1.2q+13
but I didn't know where to go from here in order to find 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! Let me try to help you out and hopefully make it a little more entertaining along the way!
To find the minimum marginal cost, you're correct in taking the derivative of the total cost function. However, you also need to set the derivative equal to zero and solve for q. Let's do that:
0.03q^2 - 1.2q + 13 = 0
Now, you have a quadratic equation that you can solve to find the value(s) of q at which the marginal cost is at a minimum. You can use the quadratic formula or factoring to solve it. Plugging in those values, you can find the corresponding minimum marginal cost.
Once you find the value of q that minimizes the marginal cost, you can substitute it back into the total cost equation to find the total cost per item at that production level.
The practical interpretation of the minimum marginal cost is that it represents the additional cost of producing one extra item at that production level. So, when the marginal cost is at its lowest, it means that the cost of producing an additional item is minimized.
Now, to find the production level at which the average cost is at a minimum, you need to find the value of q that minimizes the average cost function. You can use a similar approach by taking the derivative of the average cost function, setting it equal to zero, and solving for q. That value will give you the production level at which the average cost is minimized. Substituting that value back into the average cost function will give you the lowest average cost.
Finally, to find the marginal cost at q = 30, you can plug that value into the derivative equation you found earlier: 0.03(30)^2 - 1.2(30) + 13.
I hope this helps! Don't hesitate to ask if you have any more questions.
No problem! I'm here to help. Let's break down your questions step-by-step:
1. What is the minimum marginal cost?
To find the minimum of the marginal cost function, we need to find the value of q where the derivative of the total cost function is equal to zero. You correctly calculated the derivative of the total cost function as: mc(q) = 0.03q^2 - 1.2q + 13.
To find the minimum of mc(q), we can set the derivative equal to zero and solve for q:
0.03q^2 - 1.2q + 13 = 0
Unfortunately, this quadratic equation does not yield simple integer solutions. We can find the minimum value of the marginal cost using calculus techniques, such as finding the vertex of the quadratic function.
2. What is the practical interpretation of this result?
The practical interpretation of the result is that the marginal cost is at a minimum when the production level is at a certain point, q. The minimum marginal cost signifies the point at which the cost of producing an additional unit is the lowest.
3. At what production level is the average cost a minimum? What is the lowest average cost?
To find the production level at which the average cost is minimized, we need to find the minimum point of the average cost function. The average cost function is given as: a(q) = 0.01q^2 - 0.6q + 13.
To find the minimum of a(q), we can set the derivative equal to zero and solve for q:
0.02q - 0.6 = 0
q = 0.6 / 0.02
q = 30
The production level at which the average cost is minimized is q = 30. To find the lowest average cost, substitute this value back into the average cost function:
a(30) = 0.01(30)^2 - 0.6(30) + 13
Compute the answer to find the lowest average cost.
4. Compute the marginal cost at q = 30.
To compute the marginal cost at q = 30, substitute this value into the marginal cost function you derived earlier:
mc(30) = 0.03(30)^2 - 1.2(30) + 13
Calculate the value to find the marginal cost at q = 30.
I hope this helps! Let me know if you have any further questions.
No problem at all! I'm here to help you understand. Let's break down your questions step by step.
First, let's find the minimum marginal cost. You correctly found the derivative of the total cost function, which is equal to the marginal cost. The derivative you found, .03q^2 - 1.2q + 13, represents the rate at which the total cost changes with respect to the quantity produced.
To find the minimum, we need to set the derivative equal to zero and solve for q:
.03q^2 - 1.2q + 13 = 0
This is a quadratic equation. You can solve it by factoring or using the quadratic formula. Once you solve for q, you will find the value(s) that make the derivative zero. These values correspond to the minimum marginal cost.
Once you find the value(s) of q, substitute them back into the derivative to find the minimum marginal cost.
Now, let's move on to the practical interpretation of this result. The marginal cost represents the additional cost of producing one more unit of the good. When the marginal cost is at a minimum, it means that the cost of producing one additional item is the lowest it can be.
To determine the total cost per item at this minimum point, you need to substitute the value(s) of q that you found into the total cost function. This will give you the lowest average cost per item.
For the next question, you are correct. The first blank refers to the total cost, and the second blank refers to the answer you found in part a.
Finally, to find the production level at which the average cost is a minimum, you need to set the derivative of the average cost function equal to zero and solve for q. This will give you the quantity at which the average cost is minimized. Then, substitute this value into the average cost function to find the lowest average cost.
To compute the marginal cost at q = 30, substitute q = 30 into the derivative you found and compute the value. This will give you the marginal cost at that production level.
I hope this helps clarify the steps you need to take in order to find the answers to your questions. Should you have any further questions, feel free to ask!
marginal cost = .03q^2-1.2q+13
where is that parabola a minimum
If you know calculus take derivative and set to zero
otherwise find the vertex of the parabila
I will use calculus
0 = .06 q - 1.2
so min when q = 1.2/.06 = 20
then calculate
q min = .03(20)^2 -1.2(20) + 13
the practical result is that an additional item costs least when you make about 20 of them