Problem 4 - Calculus Concepts

Question

Airline Sales The function \(t\) gives the number of oneway tickets from Boston to Washington D.C. that a certain airline sells in 1 week when the price of each ticket is \(p\) dollars. Interpret the following: a. \(t(115)=1750\) b. \(t^{\prime}(115)=220\) c. \(t^{\prime}(125)=22\)

Step-by-Step Solution

Answer

a. 1750 tickets are sold at $115 each. b. Selling 220 fewer tickets per $1 price increase from $115. c. Selling 22 fewer tickets per $1 price increase from $125.

Step 1: Interpret Function Value

To interpret \(t(115) = 1750\), consider \(t(p)\) as the number of tickets sold at price \(p\). Here, \(t(115) = 1750\) means that when the price of each ticket is $115, the airline sells 1750 one-way tickets in one week.

Step 2: Interpret Derivative at a Price

The derivative \(t'(115) = 220\) indicates the rate of change in the number of tickets sold with respect to the price when the price is \(115. It means that if the price increases by \)1 from $115, approximately 220 fewer tickets will be sold per week.

Step 3: Interpret Second Derivative Information

Similarly, \(t'(125) = 22\) conveys that the rate of change in the number of tickets sold with respect to price is 22 when the price is \(125. This indicates a decrease of approximately 22 tickets for each \)1 increase in price around $125.

Key Concepts

Function Interpretation

In calculus, interpreting functions involves understanding what the function and its values represent in a real-world context. Take the function \( t(p) \), which represents the number of one-way tickets an airline sells depending on the ticket price \( p \). Thinking of it as a mapping from price to sales helps us grasp the relationship and predict outcomes. For instance, \( t(115) = 1750 \) tells us that setting the ticket price at $115 results in 1,750 tickets being sold. This interpretation is crucial as it converts abstract numbers into tangible information. It helps frame the relationship between price adjustments and sales performance for airlines or similar businesses.

• **Function values give specific outcomes at exact prices.** They allow you to predict sales efficiently for particular pricing strategies.
• **Understanding functions** aids in decision-making and assessing how variables affect results in dynamic markets such as airfare.

By breaking down what each function's value reveals, one can better understand crucial economic shifts and customer preferences.

Derivative

In mathematics, a derivative represents the rate at which one quantity changes with respect to another. For the function \( t(p) \), its derivative \( t'(p) \) tells us how ticket sales are expected to adjust to changes in price. Interpreting \( t'(115) = 220 \) reveals that a \(1 increase in ticket price from \)115 reduces sales by about 220 tickets per week. Hence, derivatives provide insight on how sensitive the airline's customers are to price changes.

• **Derivative as a tool** helps identify the responsiveness of sales to pricing, aiding in optimal pricing strategies.
• **Rate of change** quantifies the expected shift, ensuring that airlines can better forecast and strategize around customer behavior.

Derivatives thus act as indicators of elasticity, allowing companies to anticipate the repercussions of pricing choices on sales performance.

Rate of Change

The rate of change in calculus is a key concept to understand how swiftly one quantity changes in relation to another variable. It's the numerical representation of a derivative. In this context, for \( t'(125) = 22 \), it signifies that at the price of \(125, increasing the price by \)1 would lead to a loss of about 22 tickets per week.
This means:

• **Elasticity in demand:** The small decline in tickets implies less sensitivity to price changes at \(125, compared to \)115 where the change was more significant (220 tickets).
• **Business implications:** Understanding these patterns helps adjust pricing strategies to either mitigate losses or maximize sales, depending on the price point.

Recognizing how rates of change differ across price levels enables more effective pricing policies that align with market demand and customer purchase behavior. It's invaluable for fine-tuning offerings to ensure competitive advantage and meeting revenue targets.