Problem 6 - Calculus Concepts

Question

a. Write the units of measure for the input and output variables of the demand function. b. Write a sentence of interpretation for each point given. The function \(D\) gives the number of single-person aircraft ordered when a single-person aircraft sells for \(p\) thousand dollars. (64,560,000)\(;(320,2000)\)

Step-by-Step Solution

Answer

Input is in thousand dollars; output is quantities ordered. (64, 560,000): at $64,000, 560,000 orders. (320, 2000): at $320,000, 2000 orders.

Step 1: Identify the Units of Measure for Input and Output Variables

In the demand function \( D \), the input variable \( p \) represents the price of a single-person aircraft in units of thousand dollars. The output variable \( D(p) \) signifies the number of aircraft ordered, which is unitless as it counts entities (aircrafts).

Step 2: Interpret the First Point (64, 560,000)

The point (64, 560,000) can be interpreted as follows: when the price of a single-person aircraft is \( \$64,000 \) (since \( p = 64 \) thousand dollars), 560,000 aircraft are ordered.

Step 3: Interpret the Second Point (320, 2000)

The second point (320, 2000) signifies that when the price of a single-person aircraft is \( \$320,000 \) (since \( p = 320 \) thousand dollars), 2000 aircraft are ordered.

Key Concepts

Input Variable

When dealing with a demand function, understanding the role of the input variable is crucial. Here, the input variable is denoted by \( p \), representing the price of a single-person aircraft in units of thousand dollars. In simpler terms, \( p \) is the amount of money needed to purchase one aircraft, expressed in thousands. This means that if \( p = 64 \), it actually means \( \\(64,000 \). The concept of using thousands helps in simplifying large numbers, making it easier to handle figures related to pricing in a demand model.
Remember: Without knowing the scale of the money (thousands here), the interpretation could be misleading.

• \( p = 64 \) means a price of \( \\)64,000 \).
• \( p = 320 \) means a price of \( \$320,000 \).

Output Variable

The output variable in the demand function \( D(p) \) signifies the number of single-person aircraft ordered. As an output of the function, it tells us what the market demand is based on a given price point for the aircraft. What's essential to note here is that \( D(p) \) is unitless regarding traditional units like meters or kilowatts.
It's a straightforward count of items — in this case, the number of aircraft. It reflects the quantity that consumers are willing or able to purchase at a particular price level. This output is pivotal for manufacturers to understand how price fluctuations might affect demand.

• At \( p = 64 \) or \( \\(64,000 \), \( D(64) = 560,000 \) aircraft are ordered.
• At \( p = 320 \) or \( \\)320,000 \), \( D(320) = 2000 \) aircraft are ordered.

Interpretation of Points

Interpreting specific points on the demand function helps to clarify the relationship between price and order quantity. For the point \((64, 560,000)\), this tells us that when the aircraft is priced at \( \\(64,000 \), a massive number of 560,000 aircraft are ordered. This might suggest a high demand when prices are relatively low.
On the other hand, the point \((320, 2000)\) provides a different insight. Here, when the price rises to \( \\)320,000 \), only 2000 aircraft are ordered. The stark contrast in these two scenarios highlights the sensitivity of demand to price changes.
Let's break this down further:

• As the price increases from \( \\(64,000 \) to \( \\)320,000 \), the demand decreases dramatically from 560,000 to just 2000 aircraft.
• This suggests a typical inverse relationship in demand functions: higher prices lead to lesser demand, assuming all other factors remain constant.

Understanding these points can be valuable for predicting how future pricing strategies might affect demand.