Problem 29 - Calculus Concepts

Question

The demand for tickets to a children's museum can be modeled as $$ D(p)=0.03 p^{2}-1.6 p+21 \text { thousand tickets } $$ where \(p\) is the market price. a. What is the price elasticity of demand at a market price of \(\$ 15\) per ticket? b. Is demand elastic or inelastic at \(\$ 15\) per ticket? c. Explain in context what elasticity (or inelasticity) at \(\$ 15\) per ticket means.

Step-by-Step Solution

Answer

a. Elasticity at \(p = 15\) is \(-2.8\). b. Demand is elastic. c. Demand is responsive to price changes, increasing 2.8% for each 1% decrease in price.

Step 1: Differentiate the Demand Function

The first step is to find the derivative of the demand function \(D(p) = 0.03p^2 - 1.6p + 21\) with respect to \(p\). This derivative, \(D'(p)\), represents the rate of change of demand with respect to price. Using the power rule, we have:\[ D'(p) = 0.06p - 1.6 \]

Step 2: Calculate Demand at $p = 15$

Next, substitute \(p = 15\) into the original demand function to find \(D(15)\), the quantity demanded at \(p = 15\):\[ D(15) = 0.03(15)^2 - 1.6(15) + 21 \]Calculating this gives:\[ D(15) = 0.03 imes 225 - 24 + 21 = 6.75 - 24 + 21 = 3.75 \]Thus, the quantity demanded is 3.75 thousand tickets.

Step 3: Calculate Derivative at $p = 15$

Substitute \(p = 15\) into the derivative \(D'(p)\) to find the rate of change of demand when \(p = 15\):\[ D'(15) = 0.06 imes 15 - 1.6 \]Calculating this gives:\[ D'(15) = 0.9 - 1.6 = -0.7 \]

Step 4: Calculate Price Elasticity of Demand

Price elasticity of demand \(E(p)\) can be calculated using the formula:\[ E(p) = \frac{D'(p) \times p}{D(p)} \]Substitute all known values (\(D'(15) = -0.7, D(15) = 3.75, p = 15\))\[ E(15) = \frac{-0.7 \times 15}{3.75} \]\[ E(15) = \frac{-10.5}{3.75} = -2.8 \]

Step 5: Determine Elasticity

Demand is said to be elastic if the absolute value of price elasticity is greater than 1, and inelastic if it is less than 1. In this case, \(|E(15)| = 2.8\), which is greater than 1, indicating that demand is elastic at \(p = 15\).

Step 6: Interpret Elasticity

Elasticity at \(p = 15\) means that the demand for tickets is responsive to changes in price. More specifically, since the elasticity \(-2.8\) is less than \(-1\), a 1% decrease in price will result in an approximately 2.8% increase in the quantity of tickets demanded.

Key Concepts

Demand Function

A demand function represents how the quantity of a product demanded by consumers changes with the market price. In our example, the demand for children's museum tickets is illustrated by the function: \[ D(p) = 0.03p^2 - 1.6p + 21 \] Here, \( p \) is the market price of a ticket. The demand function allows businesses to predict how many tickets might be sold at various price points.

• The quadratic nature of the function implies that changes in price will have varied effects on demand.

This mathematical expression is valuable because it informs us about consumer behavior with respect to price changes, helping in strategic price setting.

Derivative

The derivative of a function shows how one quantity changes in response to a change in another quantity. In this context, the derivative of the demand function \( D'(p) = 0.06p - 1.6 \) tells us how sensitive the demand for tickets is to changes in price.

• By calculating the derivative, we know the rate of change of demand when the price changes.
• For instance, at \(p = 15\), \(D'(15) = -0.7\), indicating that with each dollar increase in price, the quantity demanded decreases by 0.7 thousand tickets.

Knowing the derivative helps businesses assess and predict market demand trends based on varying prices.

Elasticity Interpretation

Price elasticity of demand is a measure that shows how much the quantity demanded responds to changes in price. It is calculated using the formula: \[ E(p) = \frac{D'(p) \times p}{D(p)} \] At \(p = 15\), we found \( E(15) = -2.8 \), meaning the demand is elastic. This indicates that demand changes significantly with price changes.

• In this scenario, since \(|E(15)| = 2.8\), a 1% decrease in price increases the demand by roughly 2.8%.
• Elastic demand suggests consumers are sensitive to price changes, influencing pricing strategies.

Understanding elasticity helps businesses adjust prices to maximize revenue, ensuring it's neither too high to daunt customers nor too low to diminish profits.

Market Price

Market price refers to the current price at which a good or service can be bought or sold. In our case, the market price of museum tickets is \(\$ 15\).

• This price is crucial as it directly influences demand, derived from consumer purchasing habits and willingness to pay.
• Determining the right market price involves analyzing demand functions and elasticity.

A well-set market price helps achieve the balance between maximizing sales volume and achieving optimal revenue and profit goals. Understanding the role of market price in demand dynamics is vital for setting strategic pricing.