Problem 8 - Applied Calculus

Question

An amusement park charges an admission fee of $$\$ 7$$ per person as well as an additional $$\$ 1.50$$ for each ride. (a) For one visitor, find the park's total revenue \(R(n)\) as a function of the number of rides, \(n\), taken. (b) Find \(R(2)\) and \(R(8)\) and interpret your answers in terms of amusement park fees.

Step-by-Step Solution

Answer

\( R(n) = 7 + 1.50n \). For 2 rides, \( R(2) = \$10 \); for 8 rides, \( R(8) = \$19 \).

Step 1: Understanding the Problem

We need to find the total revenue, \( R(n) \), for one visitor based on the number of rides, \( n \), taken. The park charges a flat fee of \\(7 per visitor plus \\)1.50 for each ride.

Step 2: Setting up the Revenue Function

The total revenue for \( n \) rides can be expressed as a function, \( R(n) = 7 + 1.50n \). Here, \\(7 represents the admission fee and \\)1.50n represents the cost of \( n \) rides.

Step 3: Calculate R(2)

Find \( R(2) \) by substituting \( n = 2 \) into the revenue function: \[ R(2) = 7 + 1.50 \times 2 = 7 + 3 = 10. \] So, for 2 rides, the total revenue is \$10.

Step 4: Calculate R(8)

Find \( R(8) \) by substituting \( n = 8 \) into the revenue function: \[ R(8) = 7 + 1.50 \times 8 = 7 + 12 = 19. \] So, for 8 rides, the total revenue is \$19.

Key Concepts

Revenue Function

When trying to understand a revenue function in applied calculus, it's important to first grasp what "revenue" itself means. In a business context, revenue is the money generated from normal business operations. This might include product sales, service fees, or other income streams.
The revenue function is a mathematical expression that helps calculate total revenue based on certain variables. In this case, our variable is the number of rides taken at an amusement park.

• The amusement park has a fixed entry fee of \(\\(7\), which is added to the total revenue irrespective of the number of rides.
• The variable cost is \(\\)1.50\) per ride, which means it increases with each additional ride the visitor takes.

The revenue function is expressed as:\[ R(n) = 7 + 1.50n \] Here, \(R(n)\) represents the total revenue, \(7\) is the admission fee, and \(1.50n\) accounts for the cost of \(n\) rides taken by a visitor.

Linear Equations

Linear equations are a fundamental part of algebra and calculus, used to describe relationships between variables with a constant rate of change. In the context of this amusement park problem, we have a linear equation for calculating revenue.
A linear equation is structured in the form \(y = mx + b\). - In our example, \(R(n) = 7 + 1.50n\), where:

• \(R(n)\) replaces \(y\)
• \(1.50\) is the slope \(m\), indicating the cost increase per ride
• \(7\) is the \(b\) or y-intercept, representing the fixed fee

The slope tells us how quickly revenue increases with additional rides, while the y-intercept shows the base charge without rides. The equation remains linear because the cost increases by a fixed amount with each ride, making the graph a straight line. Unlike quadratic or exponential equations, linear equations do not have curves or acceleration/deceleration.

Arithmetic Operations

Arithmetic operations refer to basic calculations such as addition, subtraction, multiplication, and division. These operations form the backbone of solving equations and understanding functions in mathematics.
When working with our revenue function \(R(n) = 7 + 1.50n\), we primarily use addition and multiplication.

• **Addition:** We start by adding the fixed admission fee (\(\\(7\)) to the total cost of rides.
• **Multiplication:** We calculate the total cost of the rides by multiplying the number of rides \(n\) by \(\\)1.50\) per ride.

Calculating specific values like \(R(2)\) and \(R(8)\) involves simple substitution and arithmetic:- For \(R(2)\): Calculate \(7 + 1.50 \times 2\) which equals \(7 + 3 = 10\). - For \(R(8)\): Calculate \(7 + 1.50 \times 8\) which equals \(7 + 12 = 19\). These arithmetic operations transform abstract functions into specific, interpretable figures.