Question

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 \(50 t h\) product is \(\$ 6.20,\) it costs \(\$ 6.20\) to increase production from 49 to 50 units of output. Suppose the marginal cost \(C\) (in dollars) to produce \(x\) thousand digital music players is given by the function $$ C(x)=x^{2}-140 x+7400 $$ (a) How many players should be produced to minimize the marginal cost? (b) What is the minimum marginal cost?

Short answer

70 thousand players should be produced. The minimum marginal cost is \$2500.

Step-by-step solution

Identify the Problem

We are given a marginal cost function, and we need to determine the number of units produced that will minimize this cost, as well as the minimum marginal cost.

Write Down the Given Function

The marginal cost function is given as: \[ C(x) = x^2 - 140x + 7400 \]

Find the Derivative of the Function

To find the number of players that should be produced to minimize the marginal cost, first, find the derivative of the function. The derivative of \( C(x) \) is: \[ C'(x) = \frac{d}{dx}(x^2 - 140x + 7400) = 2x - 140 \]

Set the Derivative to Zero

Set the derivative equal to zero to find the critical points: \[ 2x - 140 = 0 \]

Solve for x

Solving for \(x\): \[ 2x = 140 \] \[ x = 70 \] The number of players that should be produced to minimize the marginal cost is 70 thousand.

Calculate the Minimum Marginal Cost

Substitute \( x = 70 \) back into the original marginal cost function to find the minimum marginal cost: \[ C(70) = 70^2 - 140(70) + 7400 \] \[ C(70) = 4900 - 9800 + 7400 \] \[ C(70) = 2500 \] The minimum marginal cost is \$2500.

Key concepts

Marginal Cost Function

The marginal cost of a product represents the cost to produce one additional unit. It helps businesses understand the cost implications of increasing production.
This concept is essential in economics and business because it sheds light on how production costs change with each additional unit.
Let's take the given function in the exercise, which states the marginal cost (in dollars) to produce \(x\) thousand units of digital music players:
C(x) = x^{2} - 140x + 7400. This quadratic function expresses how costs evolve with respect to the number of units produced.
By analyzing this function, one can determine not only the marginal cost at specific production levels but also identify the optimal production level where costs are minimized.

Derivative

The derivative of a function measures how the function's output changes as its input changes. In the context of a marginal cost function, the derivative represents the rate of change of marginal cost.
For the given function \(C(x) = x^{2} - 140x + 7400\), we find its derivative by applying basic rules of differentiation:
C'(x) = \frac{d}{dx} (x^{2} - 140x + 7400) = 2x - 140. This derivative, C'(x), tells us how quickly the marginal cost is changing at any given production level \(x\).
By studying the derivative, we can pinpoint where the marginal cost stops increasing and starts decreasing—marking an important aspect in the optimization process.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. In the context of our marginal cost function, critical points help identify the production levels that could potentially minimize or maximize the cost.
For the derived function C'(x) = 2x - 140, we set the derivative equal to zero to find the critical points:
2x - 140 = 0. Solving for \(x\), we get:
x = 70. This critical point means that producing 70 thousand units may minimize the marginal cost.
Further analysis, by substituting this value back into the original function, verifies if it indeed corresponds to a minimal cost.

Optimization Problem

An optimization problem involves finding the best solution from all feasible solutions. In this exercise, we aim to find the production level that minimizes the marginal cost.
Once we identify the critical point, 70 thousand units, we substitute back to the original marginal cost function to find the minimum cost:
C(70) = 70^{2} - 140(70) + 7400. Simplifying, we have:
C(70) = 4900 - 9800 + 7400. This calculation results in:
C(70) = 2500. Thus, the minimum marginal cost is $2500.
Solving these types of optimization problems assists businesses in making efficient production and cost-related decisions, ensuring optimal allocation of resources.