Problem 54
Question
Tongue-Tied Sauces, Inc., finds that the cost, in dollars, of producing \(x\) bottles of barbecue sauce is given by \(C(x)=375+0.75 x^{3 / 4}\). Find the rate at which average cost is changing when 81 bottles of barbecue sauce have been produced.
Step-by-Step Solution
Verified Answer
The average cost is decreasing at a rate of \(-\frac{1}{152}\) dollars per bottle when 81 bottles are produced.
1Step 1: Define the Average Cost Function
The average cost function, denoted as \( AC(x) \), is found by dividing the total cost function \( C(x) \) by \( x \), the number of bottles produced. Thus, \( AC(x) = \frac{C(x)}{x} = \frac{375 + 0.75 x^{3/4}}{x} \).
2Step 2: Calculate the Derivative of the Average Cost Function
To find the rate of change of the average cost, we need the derivative of the average cost function \( AC(x) \). Simplify \( AC(x) \) first: \[ AC(x) = \frac{375}{x} + 0.75 x^{-1/4} \]. Then differentiate: \[ AC'(x) = -\frac{375}{x^2} - \frac{3}{16} x^{-5/4} \].
3Step 3: Evaluate the Derivative at 81 Bottles
To find the rate at which the average cost is changing when 81 bottles are produced, substitute \( x = 81 \) into \( AC'(x) \). Calculate each part carefully: \[ AC'(81) = -\frac{375}{81^2} - \frac{3}{16} \times 81^{-5/4} \].
4Step 4: Perform Calculations
Compute \( 81^2 = 6561 \), thus \(-\frac{375}{6561}\). Next, find \( 81^{-5/4} \): break it into steps, \( 81^{1/4} = 3 \) since 81 is \( 3^4 \), then \( 81^{-5/4} = 3^{-5} = \frac{1}{243} \). Compute \(-\frac{3}{16} \times \frac{1}{243} = -\frac{1}{1296} \). The derivative evaluation results in \[ AC'(81) = -\frac{375}{6561} - \frac{1}{1296} \].
5Step 5: Final Calculation and Interpretation
Combine \(-\frac{375}{6561}\) and \(-\frac{1}{1296}\) under a common denominator. The least common multiple of 6561 and 1296 is 10368. Convert both fractions: \(-\frac{375}{6561} = -\frac{60}{10368}\) and \(-\frac{1}{1296} = -\frac{8}{10368}\). Thus, \[ AC'(81) = -\frac{68}{10368} = -\frac{1}{152} \]. This means the average cost is decreasing at a rate of \( \frac{1}{152} \) dollars per bottle when 81 bottles are produced.
Key Concepts
Derivative of Average CostRate of ChangeAverage Cost Function
Derivative of Average Cost
Understanding the concept of the derivative of the average cost can be incredibly helpful in economics and calculus. The derivative of a function essentially tells us how the function is changing at any given point. In the context of the average cost function, the derivative, denoted here as\( AC'(x) \), measures the rate at which the average cost per bottle is changing with respect to the bottle production count,\( x \).
To find the derivative of the average cost, first, you must simplify the average cost function\( AC(x) \). In our case \( AC(x) = \frac{375}{x} + 0.75 x^{-1/4} \). Taking the derivative requires knowledge of basic derivative rules:
Applying these, the derivative, or the rate of change of the average cost with respect to \( x \), becomes \(-\frac{375}{x^2} - \frac{3}{16} x^{-5/4} \). This derivative can inform businesses about how their average cost will behave as production scales up or down.
To find the derivative of the average cost, first, you must simplify the average cost function\( AC(x) \). In our case \( AC(x) = \frac{375}{x} + 0.75 x^{-1/4} \). Taking the derivative requires knowledge of basic derivative rules:
- The derivative of \( x^{-n} \) is \(-n x^{-n-1} \).
- The derivative of a constant with respect to a variable is zero.
Applying these, the derivative, or the rate of change of the average cost with respect to \( x \), becomes \(-\frac{375}{x^2} - \frac{3}{16} x^{-5/4} \). This derivative can inform businesses about how their average cost will behave as production scales up or down.
Rate of Change
The concept of rate of change is essential for understanding how variables interact over time or quantity.
In the context of cost functions, the rate of change of the average cost, found by the derivative\( AC'(x) \), directly shows us how efficient cost management can be according to production levels.
When you calculate \( AC'(81) \), you are determining the rate at which the average cost changes at the specific level of production of 81 bottles.
Mathematically, when\( AC'(x) \) results in negative values, it means that as you produce more items, the average cost per item is decreasing, signaling potential economies of scale.
In our problem, \( AC'(81) = -\frac{1}{152} \), indicating that producing one additional bottle at the 81-bottle level decreases the average cost by about \(\frac{1}{152} \) dollar per bottle.
In the context of cost functions, the rate of change of the average cost, found by the derivative\( AC'(x) \), directly shows us how efficient cost management can be according to production levels.
When you calculate \( AC'(81) \), you are determining the rate at which the average cost changes at the specific level of production of 81 bottles.
Mathematically, when\( AC'(x) \) results in negative values, it means that as you produce more items, the average cost per item is decreasing, signaling potential economies of scale.
In our problem, \( AC'(81) = -\frac{1}{152} \), indicating that producing one additional bottle at the 81-bottle level decreases the average cost by about \(\frac{1}{152} \) dollar per bottle.
Average Cost Function
The average cost function is a crucial tool for businesses to determine how costs distribute over units of production. It helps in setting pricing strategies and managing budgets.
The average cost (\( AC \)) is calculated by dividing the total cost \( C(x) \) of producing \( x \) items by \( x \) itself.
In our equation, the total cost is given by \( C(x) = 375 + 0.75 x^{3/4} \), and thus the average cost function is \( AC(x) = \frac{375 + 0.75 x^{3/4}}{x} \).
By understanding how these components of the average cost behave with varying \( x \), companies can better strategize production levels and pricing to maximize profits.
The average cost (\( AC \)) is calculated by dividing the total cost \( C(x) \) of producing \( x \) items by \( x \) itself.
In our equation, the total cost is given by \( C(x) = 375 + 0.75 x^{3/4} \), and thus the average cost function is \( AC(x) = \frac{375 + 0.75 x^{3/4}}{x} \).
- The term \( \frac{375}{x} \) indicates the fixed costs distributed per item.
- The term \( 0.75 x^{-1/4} \) reflects the variable part of the cost that changes with production, influenced by complexity or efficiency of production processes.
By understanding how these components of the average cost behave with varying \( x \), companies can better strategize production levels and pricing to maximize profits.
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