Problem 161
Question
161\. The price \(p\) (in dollars) and the demand \(x\) for a certain digital clock radio is given by the price-demand function \(p=10-0.001 x\). a. Find the revenue function \(R(x)\). b. Find the marginal revenue function. c. Find the marginal revenue at \(x=2000\) and 5000 .
Step-by-Step Solution
Verified Answer
a. \( R(x) = 10x - 0.001x^2 \)
b. \( R'(x) = 10 - 0.002x \)
c. At \( x = 2000, R'(2000) = 6; x = 5000, R'(5000) = 0 \)
1Step 1: Understand the Relationship between Revenue, Price, and Demand
Revenue is calculated by multiplying the price at which a product is sold by the quantity (demand) of products sold. In this case, we are given a price-demand function, which gives the price as a function of demand: \( p(x) = 10 - 0.001x \). Therefore, the revenue function \( R(x) \) can be obtained by multiplying \( p(x) \) by \( x \).
2Step 2: Derive the Revenue Function
To derive the revenue function, multiply the price function by the demand: \( R(x) = p(x) \times x = (10 - 0.001x) \times x \). Simplifying, we get:\[ R(x) = 10x - 0.001x^2 \].
3Step 3: Find the Marginal Revenue Function
The marginal revenue function is the derivative of the revenue function with respect to the demand \( x \). To find this, differentiate \( R(x) = 10x - 0.001x^2 \): \[ R'(x) = \frac{d}{dx}(10x - 0.001x^2) = 10 - 0.002x \].
4Step 4: Calculate Marginal Revenue at Specific Demand Levels
To find the marginal revenue at \( x = 2000 \), substitute \( x = 2000 \) into the marginal revenue function: \[ R'(2000) = 10 - 0.002(2000) = 10 - 4 = 6 \].For \( x = 5000 \), substitute \( x = 5000 \): \[ R'(5000) = 10 - 0.002(5000) = 10 - 10 = 0 \].
Key Concepts
Marginal RevenuePrice-Demand FunctionDerivative of Revenue Function
Marginal Revenue
Marginal revenue is a concept frequently used in economics and business, referring to the additional income generated from selling one more unit of a good or service. The marginal revenue can inform businesses how changes in output levels impact their revenue.
To compute marginal revenue, we look at the revenue function, which represents total revenue as a function of the quantity of goods sold. By taking the derivative of the revenue function with respect to quantity, we derive the marginal revenue function. This derivative essentially captures the rate of change of revenue relative to changes in output.
In our exercise, the marginal revenue function was found by differentiating the revenue function, \[ R(x) = 10x - 0.001x^2 \], resulting in \[ R'(x) = 10 - 0.002x \].This derivative gives us a formula that calculates the change in revenue for a slight increase in demand. Thus, businesses can use this marginal revenue to make pricing decisions and optimize their output levels.
To compute marginal revenue, we look at the revenue function, which represents total revenue as a function of the quantity of goods sold. By taking the derivative of the revenue function with respect to quantity, we derive the marginal revenue function. This derivative essentially captures the rate of change of revenue relative to changes in output.
In our exercise, the marginal revenue function was found by differentiating the revenue function, \[ R(x) = 10x - 0.001x^2 \], resulting in \[ R'(x) = 10 - 0.002x \].This derivative gives us a formula that calculates the change in revenue for a slight increase in demand. Thus, businesses can use this marginal revenue to make pricing decisions and optimize their output levels.
Price-Demand Function
The price-demand function is a fundamental concept in economics that describes how price levels are determined by the demand for goods or services. It is often represented as a function where price \( p \) is expressed in terms of quantity demanded \( x \).
Understanding this function is critical because it allows firms to predict the price point at which they can expect different quantities of sales. This insight is valuable for setting prices to achieve specific financial targets, whether aiming for higher revenues or managing inventories effectively.
- This function helps illustrate how demand varies with changes in price and can be graphed to show demand elasticity.
- In the provided problem, the price-demand function given is \( p(x) = 10 - 0.001x \),indicating that the price decreases as demand increases.
Understanding this function is critical because it allows firms to predict the price point at which they can expect different quantities of sales. This insight is valuable for setting prices to achieve specific financial targets, whether aiming for higher revenues or managing inventories effectively.
Derivative of Revenue Function
The derivative of a revenue function is a potent tool that offers critical insights into how revenue changes with varying levels of demand. To derive insights regarding how incremental changes in demand affect overall revenue, firms utilize calculus.
In practical terms, if we have a revenue function \( R(x) \), the derivative \( R'(x) \) describes how a small change in the number of goods sold impacts total revenue. By performing calculus on the revenue function \[ R(x) = 10x - 0.001x^2 \],we obtain \[ R'(x) = 10 - 0.002x \].This newly formed derivative function provides us with a roadmap to understand incremental revenue changes.
Such analysis is crucial in decision-making and strategic planning for businesses as it allows them to determine the optimal level of production and pricing strategy that maximizes profitability.
In practical terms, if we have a revenue function \( R(x) \), the derivative \( R'(x) \) describes how a small change in the number of goods sold impacts total revenue. By performing calculus on the revenue function \[ R(x) = 10x - 0.001x^2 \],we obtain \[ R'(x) = 10 - 0.002x \].This newly formed derivative function provides us with a roadmap to understand incremental revenue changes.
Such analysis is crucial in decision-making and strategic planning for businesses as it allows them to determine the optimal level of production and pricing strategy that maximizes profitability.
Other exercises in this chapter
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