Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The total cost \(C(x)\) (in dollars) incurred by Aloha Company in manufacturing \(x\) surfboards a day is given by $$ C(x)=-10 x^{2}+300 x+130 \quad(0 \leq x \leq 15) $$ a. Find \(C^{\prime}(x)\). b. What is the rate of change of the total cost when the level of production is ten surfboards a day?

Step-by-Step Solution

Answer

a. The derivative of the cost function is: \( C'(x) = -20x + 300 \). b. The rate of change of the total cost when the level of production is ten surfboards a day is \(100\) dollars per surfboard.

Step 1: Compute the derivative, \(C'(x)\)

To find the rate of change of the total cost, we need to find the derivative of the cost function \(C(x)\) with respect to the number of surfboards \(x\). The cost function is given by: $$ C(x) = -10x^2 + 300x + 130. $$ Differentiating with respect to \(x\), we get: $$ C'(x) = \frac{d}{dx}(-10x^2 + 300x + 130). $$ Using the power rule for each term, we have: $$ C'(x) = -20x + 300. $$ For part a, the derivative of the cost function is: \( C'(x) = -20x + 300 \).

Step 2: Compute the rate of change at ten surfboards a day

Now we need to compute the rate of change of the total cost when the level of production is ten surfboards a day. This means we need to find the value of \(C'(x)\) at \(x = 10\). Plugging \(x = 10\) into our expression for the derivative, we get: $$ C'(10) = -20(10) + 300. $$ Simplifying, we find: $$ C'(10) = -200 + 300 = 100. $$ For part b, the rate of change of the total cost when the level of production is ten surfboards a day is \(100\) dollars per surfboard.

Key Concepts

Cost Function

A cost function in calculus is an important concept for understanding how costs evolve with production levels. In the provided exercise, the cost function is given as \( C(x) = -10x^2 + 300x + 130 \). Simply put, this equation describes how the total cost of manufacturing surfboards changes with the production quantity of surfboards \(x\) produced each day.

• The term \(-10x^2\) represents a quadratic component which indicates how costs change as production scale increases. It often reflects increasing costs as more surfboards are produced due to factors like resource limitations or inefficiencies.
• The linear term \(300x\) signifies the constant cost increase per unit produced. This is typically linked to direct materials, labor, and other variable costs.
• The constant \(130\) provides the fixed costs that do not change with production levels, such as rent or salaries.

Understanding a cost function like this helps companies in setting the right production levels to balance costs and maximize profitability. It offers a snapshot of how increases in production lead to changes in cost structures, serving as a critical tool for decision-making.

Derivatives

In calculus, derivatives are powerful tools used to find the rate of change of a function. When dealing with cost functions, calculating the derivative helps us understand how costs vary with each additional unit produced.

The given exercise requires computing the derivative \(C'(x)\) of the cost function \(C(x) = -10x^2 + 300x + 130\).

• The derivative \(C'(x) = \frac{d}{dx}(-10x^2 + 300x + 130)\) is found by differentiating each term separately, using rules of differentiation such as the power rule.
• For the term \(-10x^2\), the power rule \(\frac{d}{dx}(ax^n) = n \cdot ax^{n-1}\) gives us \(-20x\).
• The term \(300x\) becomes \(300\) as the derivative of \(ax\) with respect to x is \(a\).
• The constant \(130\) from the equation drops because the derivative of a constant is zero.

Combining these results yields the derivative \(C'(x) = -20x + 300\), which reflects the change in total costs with production.

Rate of Change

The rate of change in mathematics measures how one value changes with respect to another. In our exercise, it reflects how the total cost changes with the number of surfboards produced. This concept is crucial in identifying the cost implications of producing an additional surfboard.

By using the derivative \(C'(x) = -20x + 300\), we can assess the rate of change in costs at different production levels.

• In the original problem, evaluating the rate at 10 surfboards involves substituting \(x = 10\) into \(C'(x)\), resulting in \(C'(10) = -200 + 300 = 100\).
• Here, the rate of change at 10 units is \(100\), meaning for each additional surfboard produced at this level, the cost increases by \(100\) dollars.

Understanding the rate of change helps businesses optimize production, revealing whether it's more advantageous to produce more units or hold back based on rising or falling costs.