Problem 56 - Calculus and its applications

Question

Tongue-Tied Sauces, Inc., finds that the revenue, in dollars, from the sale of \(x\) bottles of barbecue sauce is given by \(R(x)=7.5 x^{0.7}\). Find the rate at which average revenue is changing when 81 bottles of barbecue sauce have been produced and sold.

Step-by-Step Solution

Answer

The rate at which average revenue is changing is approximately -0.0059 dollars per bottle for 81 bottles produced.

Step 1: Understand the Problem

We need to find the rate at which the average revenue is changing when 81 bottles are produced. The revenue function is given by \( R(x) = 7.5x^{0.7} \). The average revenue function is given by \( A(x) = \frac{R(x)}{x} \). We will need to calculate the derivative of this average revenue.

Step 2: Define the Average Revenue

Average revenue \( A(x) \) is calculated by dividing the total revenue \( R(x) \) by the number of bottles \( x \). So, we have:\[ A(x) = \frac{R(x)}{x} = \frac{7.5x^{0.7}}{x} = 7.5x^{-0.3} \]

Step 3: Differentiate the Average Revenue Function

To find out how average revenue changes, compute the derivative of \( A(x) = 7.5x^{-0.3} \) with respect to \( x \):\[ A'(x) = \frac{d}{dx}(7.5x^{-0.3}) = 7.5 (-0.3)x^{-0.3 - 1} = -2.25x^{-1.3} \]

Step 4: Evaluate the Derivative at 81 Bottles

Now we need to find the rate of change of the average revenue when \( x = 81 \):\[ A'(81) = -2.25(81)^{-1.3} \]Calculate the value of \( A'(81) \).

Step 5: Calculate the Numerical Value

Compute the power and then the result:\( 81^{-1.3} \approx 0.0026172 \).Thus, \( A'(81) = -2.25 \times 0.0026172 \approx -0.0058884 \).

Key Concepts

Revenue Function

In the realm of business and economics, a revenue function helps us understand how a company's income behaves in relation to the number of items sold. Here, the revenue function is provided as \( R(x) = 7.5x^{0.7} \).
This function indicates the total revenue generated from selling \( x \) bottles of barbecue sauce. In practical terms:

• \( x \) represents the number of bottles sold.
• The exponent \( 0.7 \) shows a non-linear relationship, meaning revenue increases at a varying rate as more items are sold.
• The coefficient \( 7.5 \) scales the outcome, linking directly to unit pricing or market factors influencing revenue.

Understanding this function is crucial for insights into sales strategies and forecasting revenue based on varying levels of sales or production.

Average Revenue

Average revenue is simply the revenue earned per item sold. It gives businesses a clearer view of how much money each item brings in individually, instead of looking at total revenue alone. We calculate it using the formula:

• Average Revenue, \( A(x) = \frac{R(x)}{x} \)
• For our specific problem, \( A(x) = \frac{7.5x^{0.7}}{x} = 7.5x^{-0.3} \)
• This shows a per-item revenue that decays slightly (due to the \(-0.3\) exponent) as more items are sold, reflecting changes in marginal impacts at higher production levels.

This concept is vital for understanding efficiency and can highlight potential pricing strategies if average revenue trends downwards over increased sales.

Derivative

The derivative is a core concept in calculus used to find the rate at which things change. When dealing with revenue, the derivative helps determine how revenue shifts as sales volume varies.
In our exercise, we focus on the derivative of the average revenue, \( A(x) = 7.5x^{-0.3} \).

• The derivative, \( A'(x) = -2.25x^{-1.3} \), succinctly showcases how the rate of average revenue shifts.
• Here, the negative sign indicates that average revenue decreases as more bottles are sold.

Grasping this concept allows businesses to forecast and adjust their practices, ensuring the interplay between production levels and revenue is as beneficial as possible.

Rate of Change

Rate of change is a pivotal concept for understanding the dynamics in any system. In this context, it refers to how much the average revenue per bottle alters as the number of bottles sold increases.
This boils down to computing the derivative of the average revenue function at a specific point, here for \( x = 81 \).

• We found \( A'(81) = -2.25 \, (81)^{-1.3} \), yielding approximately \( -0.0058884 \).
• This result reveals a slight decline in average revenue with each additional bottle sold at this sales level.
• Understanding the rate of change enables proactive adjustments in strategies, providing early warnings if profitability per item shows a declining trend.

Overall, this knowledge allows companies to align production and sales targets with desired revenue outcomes effectively.