Problem 16 - Applied Calculus

Question

An amusement park charges an admission fee of $$ 21\( per person as well as an additional $$ 4.50 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) = 21 + 4.5n; R(2) = 30; R(8) = 57.

Step 1: Understand the Problem

The problem provides the cost structure of an amusement park. There is a fixed admission fee of $21 per person, plus a variable cost of $4.50 for each ride taken by the visitor. We are asked to express the total cost (revenue for the park from one visitor) as a function of the number of rides.

Step 2: Formulate the Revenue Function

To find the park's total revenue function \( R(n) \), we sum the fixed admission fee and the cost depending on the number of rides. The function can be expressed as:\[ R(n) = 21 + 4.5n \]where \( n \) is the number of rides taken.

Step 3: Calculate R(2)

Substitute \( n = 2 \) into the revenue function \( R(n) \):\[ R(2) = 21 + 4.5 \times 2 = 21 + 9 = 30 \]This result indicates that if a visitor takes 2 rides, the total cost (or revenue from that visitor) is $30.

Step 4: Calculate R(8)

Substitute \( n = 8 \) into the revenue function \( R(n) \):\[ R(8) = 21 + 4.5 \times 8 = 21 + 36 = 57 \]This result indicates that if a visitor takes 8 rides, the total cost (or revenue from that visitor) is $57.

Key Concepts

Admission Fee

An admission fee is a fixed charge per person to enter a location or participate in an event. It represents the initial cost a visitor must pay, regardless of any additional activities or services they choose. In our amusement park example, this fee is set at $21 per visitor. The admission fee helps the park cover various overhead expenses such as maintenance, staffing, and utilities.

• Fixed charge per person
• Covers basic access and amenities

Understanding admission fees is vital when calculating overall expenses for visitors. It forms the base cost for anyone entering the park and is a critical component of the park's revenue model.

Variable Cost

Variable costs fluctuate based on the level of activity or usage. In the context of the amusement park, it refers to the additional cost associated with each ride taken by a visitor. For our specific scenario, this cost is $4.50 per ride. Here are a few points about variable costs:

• Directly related to the activity
• Increases with more usage (more rides)

This means as the number of rides increases, so does the total cost to the visitor and the revenue for the park. Variable costs are crucial for determining profitability and understanding customer behavior. They help in formulating pricing strategies that entice customers while covering operational expenses.

Cost Function

A cost function helps in representing the total cost incurred, in this case, by a visitor at the amusement park, based on their rides. The cost function for the park can be expressed as: \[ R(n) = 21 + 4.5n \] where \( R(n) \) is the total revenue for one visitor, 21 is the fixed admission fee, and \( 4.5n \) represents the variable cost per ride. This function shows how the total cost grows as the number of rides \( n \) increases. The cost function helps the park predict revenue based on visitor behavior and aids visitors in understanding their potential expenses.

Amusement Park Economics

Amusement park economics involves understanding various factors that contribute to the overall financial workings of a park. Key components include admission fees, variable costs, and revenue functions, which together form the park's economic foundation.

• Admission fees provide initial revenue
• Variable costs adapt to visitor activity
• Cost functions help predict overall revenue

Analyzing these elements allows park managers to set pricing that is attractive to visitors yet profitable for the business. Additionally, understanding visitor spending patterns helps with staffing, marketing, and future investment decisions. Amusement park economics is all about balancing the visitor experience with financial sustainability.