The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is $6.20, it cost $6.20 to increase productionn from 49 to 50 units of output. Suppose the marginal cost C(in dollars) to produce x thousand mp3 playersis given by the function C(x)=x^2-140x+7900.
How many players should be produced to minimize the marginal cost?
What is the minimum marginal cost?
To find the number of players that should be produced to minimize the marginal cost, we need to find the value of x that corresponds to the minimum value of the function C(x).
To do this, we can take the derivative of the function C(x) and set it equal to zero, then solve for x.
So, let's find the derivative of C(x):
C'(x) = 2x - 140
Setting C'(x) = 0:
2x - 140 = 0
Solving for x:
2x = 140
x = 140 / 2
x = 70
Therefore, to minimize the marginal cost, 70 thousand mp3 players should be produced.
Now, let's find the minimum marginal cost:
To find the minimum value of C(x), we substitute the value of x = 70 into the function C(x):
C(70) = (70)^2 - 140(70) + 7900
C(70) = 4900 - 9800 + 7900
C(70) = 3000
So, the minimum marginal cost is $3000.
To minimize the marginal cost, we need to find the value of x that corresponds to the lowest point on the graph of the function C(x) = x^2 - 140x + 7900.
To find the minimum, we can use the concept of differentiation. We differentiate the function C(x) with respect to x and set it equal to zero:
C'(x) = 2x - 140
Setting C'(x) = 0, we have:
2x - 140 = 0
2x = 140
x = 70
So, to minimize the marginal cost, x should be 70.
To find the minimum marginal cost, we substitute this value back into the function C(x):
C(70) = (70)^2 - 140(70) + 7900
C(70) = 4900 - 9800 + 7900
C(70) = 3000
Therefore, the minimum marginal cost is $3000.