Problem 26 - 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 formula is \( N = -30p + 1250 \).

Step 1: Understand the Problem

We need to find a linear demand curve described by the equation of a line, such that the number of passengers per week, \( N \), depends on the price \( p \). We have two points: \((p_1 = 25, N_1 = 500)\) and \((p_2 = 20, N_2 = 650)\).

Step 2: Determine the Slope of the Demand Curve

The slope \( m \) of the line that represents the demand curve can be found using the formula for the slope between two points: \[ m = \frac{N_2 - N_1}{p_2 - p_1} \]. Substitute the known values: \[ m = \frac{650 - 500}{20 - 25} = \frac{150}{-5} = -30 \].

Step 3: Write the Equation of the Line

Now that we know the slope, we use the point-slope form of a line: \[ N - N_1 = m(p - p_1) \]. Substitute the slope \( m = -30 \) and one of the points, say \((p_1, N_1) = (25, 500)\): \[ N - 500 = -30(p - 25) \].

Step 4: Simplify the Equation

Solve for \( N \) to get the equation of the demand curve: \[ N - 500 = -30p + 750 \]. Now, move 500 to the other side: \[ N = -30p + 750 + 500 \]. Simplify: \[ N = -30p + 1250 \].

Step 5: Confirm the Solution

To confirm the solution, check that the equation holds true for both provided points. For \(p = 25\), \(N = -30(25) + 1250 = 500\). For \(p = 20\), \(N = -30(20) + 1250 = 650\). The equation satisfies both conditions.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in mathematics, specifically in coordinate geometry and calculus. It measures the steepness or inclination of a line. For a linear demand curve, the slope is crucial because it determines how much the dependent variable (in this case, the number of passengers, \(N\)) changes with a change in the independent variable (the price, \(p\)). The formula to calculate the slope \( m \) when you have two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In our boat tour problem, the slope tells us how sensitive the number of passengers is to a change in price. When the price dropped from \(25 to \)20, passengers increased from 500 to 650. The computed slope \( m = -30 \) indicates that for every $1 decrease in ticket price, 30 more passengers are expected per week. Understanding this tells us not just the rate, but also the direction of change: a negative slope means as the price goes down, the number of passengers goes up.

Point-Slope Form

The point-slope form is a way to describe a line equation using a known point on the line and the slope of the line. The standard format for the point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is a specific point on the line. This form is particularly useful when you know the slope and just one point the line passes through.
In our scenario involving the demand for boat tours, we used this form to start building the demand equation. By substituting the known point \((p_1, N_1) = (25, 500)\) and the slope \( m = -30 \) into the equation, we get \( N - 500 = -30(p - 25) \). This equation is crucial as it lays the foundation for obtaining the linear demand equation that describes the relationship between price and passenger numbers.

Demand Equation

A demand equation is an expression that shows how quantity demanded varies with price. Typically, in economics, it represents consumer behavior under different price scenarios. For linear demand curves, the equation is of the form \( Q = mP + b \), where \( Q \) is the quantity demanded, \( P \) is the price, \( m \) is the slope, and \( b \) is the y-intercept.
From the example with boat tours, after forming the point-slope equation, we further simplify to get \( N = -30p + 1250 \). This equation conveys that the number of passengers (\( N \)) changes with the ticket price (\( p \)). The intercept, 1250 in this case, suggests the hypothetical number of passengers if the price were to drop to zero. Although this scenario might be unrealistic, it provides insights when analyzing how passenger numbers start to decrease as prices increase.

Problem-Solving in Calculus

Problem-solving in calculus often involves breaking down the problem into manageable steps, using established mathematical concepts and rules. Linear equations, like the one we derived for the demand curve, are an essential part of such processes.
Our task was to determine how price influenced demand, taking calculated steps starting with recognizing there's a linear relationship. We did this by calculating the slope, employing point-slope form, and then simplifying to present a clear demand equation. Each step is vital—calculating the slope to see how change occurs, using point-slope form to structure the line, and simplifying to understand the broader implications. This systematic approach is a valuable skill that extends beyond calculus, aiding in analytical thinking and strategic problem-solving across disciplines.