Suppose a publishing company has found that the marginal cost at a level of production of x thoudsand books is given by:
C'(x)=50/sqrroot(x)
and the fixed cost is $25,000. Find the cost function C(x).
C= INT C'= INT 50/sqrtx= 100 sqrtx + F where F is the fixed cost at x=0
To find the cost function C(x), we need to integrate the marginal cost function C'(x) with respect to x.
The given marginal cost function is:
C'(x) = 50/sqrt(x)
To integrate this function, we can apply the power rule of integration. The power rule states that integrating x^k with respect to x results in (x^(k+1))/(k+1), where k is any real number except -1.
In this case, we have an exponent of -1/2 in the denominator of the marginal cost function. Adding 1 to the exponent gives us -1/2 + 1 = 1/2.
Let's integrate the function:
∫(50/sqrt(x)) dx
= ∫(50 * x^(-1/2)) dx
Applying the power rule, we get:
= 50 * (x^(1/2))/(1/2) + C
Simplifying further, we have:
= 100 * sqrt(x) + C
Now, we need to include the fixed cost, which is $25,000.
This fixed cost represents the cost at x=0.
So, we can write our cost function C(x) as:
C(x) = 100 * sqrt(x) + 25,000
Therefore, the cost function C(x) is given by 100 * sqrt(x) + 25,000.