Question

The marginal cost \(C\) (in dollars) of manufacturing \(x\) smartphones (in thousands) is given by $$ C(x)=5 x^{2}-200 x+4000 $$ (a) How many smartphones should be manufactured to minimize the marginal cost? (b) What is the minimum marginal cost?

Short answer

(a) 20, (b) 2000 dollars

Step-by-step solution

- Understanding the Problem

The goal is to find the number of smartphones that minimize the marginal cost and then to determine what that minimum marginal cost is. The given function represents the marginal cost, which is a quadratic function, so we need to find the vertex of this parabola.

- Finding the Vertex

For a quadratic function in the form of \(C(x) = ax^2 + bx + c\), the x-coordinate of the vertex, which gives the minimum or maximum value, is found using the formula \(x = -\frac{b}{2a}\). In this case, \(a = 5\), \(b = -200\), and \(c = 4000\). Substituting in, \(x = -\frac{-200}{2 \times 5} = \frac{200}{10} = 20\).

- Calculating the Minimum Marginal Cost

Now that we know the number of smartphones to minimize the cost is 20 (in thousands), we substitute \(x = 20\) back into the marginal cost function. \(C(20) = 5(20)^2 - 200(20) + 4000\). Simplifying:\(C(20) = 5(400) - 4000 + 4000 = 2000\).

Key concepts

Quadratic Functions

Quadratic functions are a type of polynomial function with the highest exponent of the variable being 2. These functions take the form \[ f(x) = ax^2 + bx + c \]where

• a is the coefficient of the squared term, where a ≠ 0
• b is the coefficient of the linear term
• c is the constant term

In the given problem, the marginal cost function is a quadratic function. Quadratic functions graph as a parabola, and the direction the parabola opens depends on the sign of a. If a is positive, the parabola opens upwards, indicating a minimum point. If a is negative, the parabola opens downwards, indicating a maximum point. This characteristic is crucial for finding the minimum or maximum value in practical problems.

Vertex Formula

The vertex of a parabola represented by the quadratic function \[ f(x) = ax^2 + bx + c \]is a critical point, which gives the maximum or minimum value. The vertex formula to find the x-coordinate of the vertex is \[ x = -\frac{b}{2a} \]Here,

• a is the coefficient of the squared term
• b is the coefficient of the linear term

In the original problem, the marginal cost function is \[ C(x) = 5x^2 - 200x + 4000 \]Using the vertex formula: \[ x = -\frac{-200}{2 \times 5} = \frac{200}{10} = 20 \]This tells us that the number of smartphones that should be manufactured to minimize the marginal cost is 20 (in thousands). The vertex formula is essential because it directly provides the value of x where the function reaches its minimum or maximum.

Minimization Problem

Minimizing a function involves finding the input value that gives the smallest output value of the function. For quadratic functions opening upwards (where a > 0), the minimum value is at the vertex. For the exercise, we focus on minimizing the marginal cost function: \[ C(x) = 5x^2 - 200x + 4000 \]Using the vertex formula, we found that the function is minimized when \[ x = 20 \]This value of x minimizes the cost since the parabola opens upwards. Plugging this x-value back into the function gives the minimum marginal cost. Evaluating \[ C(20) = 5(20)^2 - 200(20) + 4000 = 5(400) - 4000 + 4000 = 2000 \]Thus, the minimum marginal cost is 2000 dollars. This demonstrates the practical use of quadratic functions and the vertex formula in optimization problems.

Marginal Cost

Marginal cost refers to the additional cost of producing one more unit of a product. It is usually derived from the total cost function and provides insight into the efficiency of production. In mathematical terms, the marginal cost function is often modeled as a polynomial, such as a quadratic function in our exercise. The given marginal cost function: \[ C(x) = 5x^2 - 200x + 4000 \]shows how the cost per thousand smartphones changes with the number of smartphones manufactured. By finding the minimum point of this function, manufacturers can determine the optimal production level that minimizes costs. It is essential to distinguish between total cost and marginal cost:

• Total cost: The overall expense incurred to produce a given quantity of goods.
• Marginal cost: The increase in cost associated with producing one additional unit.

In optimization, understanding marginal cost helps in making decisions that enhance profit margins and operational efficiency.