Problem 30 - Applied Calculus

Question

When the price, \(p,\) charged for a boat tour was $$ 25,\( the average number of passengers per week, \)N,\( was $$500 . When the price was reduced to $$ 20,\) the average number of passengers per week increased to \(650 .\) Find a formula for the demand curve, assuming that it is linear.

Step-by-Step Solution

Answer

The demand curve equation is \( N = -30p + 1250 \).

Step 1: Understand the Problem

We need to find the equation of the line that represents the demand curve based on two price and quantity points given in the problem: (\( p = 25, N = 500 \)) and (\( p = 20, N = 650 \)).

Step 2: Set up the Equation of the Line

We assume that the demand curve is linear, so it can be expressed in the form \( N = mp + c \), where \( m \) is the slope of the line, and \( c \) is the y-intercept.

Step 3: Calculate the Slope of the Line

Use the formula for the slope of a line based on two points: \( m = \frac{N_2 - N_1}{p_2 - p_1} \). Plug in the known points: \( m = \frac{650 - 500}{20 - 25} = \frac{150}{-5} = -30 \). Thus, the slope \( m = -30 \).

Step 4: Find the Y-intercept

Using one of the points and the slope, solve for \( c \) in the equation \( N = -30p + c \). Using point (25, 500): \( 500 = -30(25) + c \). Simplifying, we get \( 500 = -750 + c \). Thus, \( c = 1250 \).

Step 5: Write the Demand Curve Equation

Substitute \( m = -30 \) and \( c = 1250 \) into the formula to get the demand curve equation: \( N = -30p + 1250 \). This equation expresses the relationship between price \( p \) and average number of passengers \( N \).

Key Concepts

Linear Equation

A linear equation is a mathematical expression that defines a straight line when plotted on a graph. In its simplest form, it's given by the formula \( y = mx + b \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope, and \( b \) is the y-intercept.

• The dependent variable depends on the value of the independent variable. Here, \( N \) (number of passengers) depends on \( p \) (price).
• The linear equation helps show how changes in \( x \) affect \( y \).

In our exercise, we model the demand curve with a linear equation \( N = mp + c \). This formula helps describe how passenger numbers change as ticket prices vary, making it easier to predict trends and make informed decisions.

Slope of a Line

In a linear equation, the slope \( m \) tells us how steep the line is. It's calculated as the change in \( y \) (vertical) divided by the change in \( x \) (horizontal) between two data points. The formula to find the slope from two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• A positive slope means the line rises as it moves to the right.
• A negative slope means the line falls as it moves to the right.

For the demand curve, the slope is \(-30\), meaning for every dollar decrease in price, the number of passengers increases by 30. It's a crucial parameter linking price changes to passenger demand.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. In our equation \( N = mx + c \), \( c \) is the y-intercept. It represents the value of \( y \) (passenger numbers) when \( x \) (price) is zero. Here’s how to determine it:Using the slope and one point, substitute them into the linear equation to solve for \( c \):For \((p, N) = (25, 500)\) and slope \(m = -30\), the equation becomes:\[500 = -30(25) + c\]Solving, we find \(c = 1250\). This tells us that if the trip were free, the passenger base would start at 1250 per week, underscoring the predicted number of people interested when price is not a factor.

Price Elasticity

Price elasticity of demand is a measure of how much the quantity demanded of a good responds to a change in the price. It's calculated using the formula:\[\text{Price Elasticity} = \frac{\% \text{ change in quantity demanded}}{\% \text{ change in price}}\]Understanding elasticity helps businesses set prices optimally:

• If elasticity is greater than 1, demand is elastic. Consumers are sensitive to price changes.
• If it's less than 1, demand is inelastic. Consumers aren't much affected by price changes.

With a calculated negative slope of \(-30\), this suggests that a decrease in price leads to a proportionately larger increase in the number of passengers. Concluding, the demand for boat tours in this case is elastic, meaning pricing strategies can greatly influence the demand.