Cost A company has found that the marginal cost (in thou-...

cost A company has found that the marginal cost (in thou- sands of dollars) to produce $x$ central air conditioning units is $$C^{\prime}(x)=\frac{60 x}{5 x^{2}+e}$$ where $x$ is the number of units produced. $\begin{array}{l}{\text { (a) Find the cost function, given that the company incurs a }} \\ {\text { fixed cost of } \$ 10,000 \text { even if no units are built. }} \\ {\text { (b) The company will seek a new source of investment income }} \\ {\text { if the cost is more than } \$ 20,000 \text { to produce } 5 \text { units. Should }} \\ {\text { they seek this new source? }}\end{array}$

cost A company has found that the marginal cost (in thou-
sands of dollars) to produce $x$ central air conditioning units is
$$C^{\prime}(x)=\frac{60 x}{5 x^{2}+e}$$
where $x$ is the number of units produced.
$\begin{array}{l}{\text { (a) Find the cost function, given that the company incurs a }} \\ {\text { fixed cost of } \$ 10,000 \text { even if no units are built. }} \\ {\text { (b) The company will seek a new source of investment income }} \\ {\text { if the cost is more than } \$ 20,000 \text { to produce } 5 \text { units. Should }} \\ {\text { they seek this new source? }}\end{array}$

Key Concepts

Cost Evaluation

Cost evaluation involves calculating the total cost for a given number of units by substituting the production level into the cost function. This is important for decision-making, such as determining whether production costs exceed a certain threshold, which can lead to strategic business decisions. It requires accurate integration and proper addition of fixed costs to ensure that the evaluation reflects the true cost incurred.

Fixed Cost

Fixed costs are costs that remain constant regardless of the level of production. These costs are incurred even when no units are produced and must be included in the total cost function as a constant term. Recognizing fixed costs is crucial for accurately assessing the total cost and for making economic decisions.

Marginal Cost Function

The marginal cost function represents the instantaneous rate of change of the total cost with respect to the production level. In economic terms, it shows the additional cost incurred by producing one more unit. Understanding the marginal cost is essential because it links the production level to the cost incurred, and it is typically used as the derivative of the total cost function.

Integration for Cost Function

Integration is the mathematical process used to recover the total cost function from the marginal cost function. By integrating the marginal cost function, you sum up all the incremental costs over a given production interval. This process involves dealing with the constant of integration, which can then be determined by additional information such as fixed costs or initial conditions.

Transcript

00:01 So for problem 21, we have a word problem that asks us to find the cost function.

00:06 Given that the company incurs the fixed cost of $10 ,000.

00:11 So right now we have c prime of x.

00:15 And the prime indicates that it is the derivative.

00:18 And we have to find c of x.

00:19 And to find c of x, we just take the integral of c prime of x.

00:25 So this would be the integral of 60x over 5x squared plus e dx.

00:43 And to do this integral, we're going to use u substitution and let the denominator equal u.

00:47 So u equals 5x squared plus e.

00:54 And if you take the derivative of both sides, you get du equals 10x.

01:02 Dx and we're going to go ahead and move the 10 to the other side so it would be d u over 10 equals x d x and as you can see in the integral we have an x dx and we have an x d x x x equal to u over 10 so we can go ahead and do the substitution so we have a 60 here and that's a constant so you can move it outside of the integral so it would be 60 and then the integral of 1 over u.

01:40 And then for the green circles, we have du over 10.

01:49 And the 10 is also a constant, so you can move that out.

01:54 So 60 divided by 10.

01:57 So 6, and then the integral of 1 over u, du.

02:04 So the integral of 1 over you is ln of u.

02:09 So it would be 6, ln of you in absolute value bars, and then don't forget the plus c since it's an indefinite integral.

02:20 And now you can't leave the answer in terms of you, so you would have to plug in what we originally let you equal to.

02:30 So it would be 6, ln, in absolute value, 5x squared, plus e, and plus c.

02:48 And we were solving for c of x.

02:55 But this is not the entire problem because the problem asks us if the fixed cost was $10 ,000.

03:03 And that means we have to let c of x equal $10 ,000...