Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function D(p)=40−2pD(p) = 40-2p , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.a. According to the model, how many DVDs can be sold in a day at a price of $10?

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 33a

Chapter 3, Problem 33a

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function $D (p) = 40 - 2 p$ , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
a. According to the model, how many DVDs can be sold in a day at a price of \$10?

Verified step by step guidance

Identify the demand function given in the problem: $ D(p) = 40 - 2p $.

Substitute the price $ p = 10 $ into the demand function.

Calculate $ D(10) = 40 - 2(10) $.

Simplify the expression to find the number of DVDs sold.

Interpret the result to understand how many DVDs can be sold at the price of \$10.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Demand Function

A demand function expresses the relationship between the price of a good and the quantity demanded by consumers. In this case, the function D(p) = 40 - 2p indicates that as the price (p) increases, the quantity of DVDs sold (D) decreases. This linear relationship helps predict sales at different price points.

Elasticity of Demand

Elasticity of demand measures how sensitive the quantity demanded is to a change in price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. Understanding elasticity helps businesses make informed pricing decisions to maximize revenue.

Substitution and Revenue Maximization

Substitution refers to how consumers may switch to alternative products as prices change. For the DVD store, if prices rise significantly, customers might choose digital streaming instead. Revenue maximization occurs when the price is set at a level where the product's demand is balanced with consumer willingness to pay, ensuring optimal sales and profit.

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Textbook Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

For what prices is the demand elastic? Inelastic?

Textbook Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

Find the elasticity function for this demand function.