Problem 55 - Calculus and its applications

Question

Preston's Leatherworks find that the revenue, in dollars, from the sale of \(x\) belts is given by \(R(x)=45 x^{9 / 10}\). Find the rate at which average revenue is changing when 175 belts have been produced and sold.

Step-by-Step Solution

Answer

The rate of change of average revenue at 175 belts is approximately -0.1146 dollars per belt.

Step 1: Define Average Revenue

The average revenue is defined as the total revenue divided by the number of units sold. Therefore, the average revenue function is given by \(AR(x) = \frac{R(x)}{x} = \frac{45x^{9/10}}{x}\).

Step 2: Simplify the Average Revenue Expression

Simplify the expression for average revenue. We have \(AR(x) = 45x^{9/10 - 1} = 45x^{-1/10}\), because \(x^{9/10}/x = x^{9/10 - 10/10} = x^{-1/10}\).

Step 3: Differentiate the Average Revenue Function

To find the rate at which the average revenue is changing, we need to differentiate \(AR(x)\) with respect to \(x\). The derivative is \(AR'(x) = \frac{d}{dx}(45x^{-1/10})\).

Step 4: Apply Power Rule to Differentiate

Using the power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\), we differentiate: \(AR'(x) = 45 \cdot \left(-\frac{1}{10}\right)x^{-1/10 - 1} = -\frac{45}{10}x^{-11/10}\).

Step 5: Simplify the Derivative

Simplify the expression for the derivative to \(AR'(x) = -4.5x^{-11/10}\).

Step 6: Evaluate the Derivative at 175 Belts

Substitute \(x = 175\) into the derivative to find the rate of change of average revenue when 175 belts are sold: \(AR'(175) = -4.5 \times 175^{-11/10}\).

Step 7: Calculate the Numerical Result

Calculate \(175^{-11/10}\) using a calculator, and then multiply by \(-4.5\) to find the rate: \(AR'(175) \approx -0.1146\).

Key Concepts

Understanding Average Revenue

Average revenue is an important concept in economics and business as it represents the revenue generated per unit. Imagine selling products, like the belts at Preston's Leatherworks. When you want to figure out the average revenue per belt sold, you simply divide the total revenue by the number of belts sold. In mathematical terms, if your revenue function is represented as \( R(x) \), where \( x \) is the number of units, the average revenue function is denoted as \( AR(x) = \frac{R(x)}{x} \). Breaking it down, if we take Preston's revenue function \( R(x) = 45x^{9/10} \), the average revenue function becomes:

\( AR(x) = \frac{45x^{9/10}}{x} \)
• \( AR(x) = 45x^{-1/10} \) after simplification.

This is how you effectively calculate how much revenue each unit, or belt in this case, is contributing on average.

The Art of Differentiation

Differentiation is a key concept in calculus that helps us understand how a function changes. When you differentiate a function, you're essentially finding its slope or the rate at which it changes. In economic terms, if you're tracking revenue or cost, the derivative tells you how quickly these are increasing or decreasing with respect to production or sales. For Preston's Leatherworks, we start with the average revenue function \( AR(x) = 45x^{-1/10} \). By differentiating it, we determine how the average revenue per belt changes with each additional belt sold. This involves calculating the derivative \( AR'(x) = \frac{d}{dx}(45x^{-1/10}) \), which gives us the rate of change at any given production level.

Applying the Power Rule

The power rule is one of the simplest yet most powerful tools in calculus for differentiation. It states that for any function \( x^n \), the derivative is \( nx^{n-1} \). This rule makes finding derivatives of polynomial expressions straightforward and efficient. For instance, in our example with Preston's Leatherworks, we apply the power rule to \( AR(x) = 45x^{-1/10} \). Differentiating using the power rule gives us:

• Multiply the exponent by the coefficient: \( 45 \times -\frac{1}{10} = -4.5 \).
• Subtract one from the exponent: \(-1/10 - 1 = -11/10 \).
• Resulting in: \( AR'(x) = -4.5x^{-11/10} \).

Finally, by evaluating this derivative at \( x = 175 \), we find the rate of change of the average revenue when 175 belts are sold. It's a practical demonstration of how the power rule and differentiation help solve real-world problems.