Problem 46 - Precalculus: Mathematical for Calculus

Question

The admission fee at an amusement park is \(\$ 1.50\) for children and \(\$ 4.00\) for adults. On a certain day, 2200 people entered the park, and the admission fees collected totaled \(\$ 5050 .\) How many children and how many adults were admitted?

Step-by-Step Solution

Answer

1500 children and 700 adults were admitted.

Step 1: Identify Variables

Let the number of children who entered the park be denoted by \( c \) and the number of adults be denoted by \( a \). We need to find the values of \( c \) and \( a \).

Step 2: Set Up Equations

Based on the information given, we have two equations:1. The total number of people: \( c + a = 2200 \)2. The total admission fees: \( 1.50c + 4.00a = 5050 \)

Step 3: Solve the First Equation for One Variable

From the first equation, solve for \( a \):\[ a = 2200 - c \]

Step 4: Substitute into the Second Equation

Substitute \( a = 2200 - c \) into the second equation:\[ 1.50c + 4.00(2200 - c) = 5050 \]

Step 5: Simplify and Solve for \( c \)

Expand and simplify the equation:\[ 1.50c + 8800 - 4.00c = 5050 \]\[ -2.50c + 8800 = 5050 \]Subtract 8800 from both sides:\[ -2.50c = 5050 - 8800 \]\[ -2.50c = -3750 \]Divide by \(-2.50\):\[ c = \frac{-3750}{-2.50} \]\[ c = 1500 \]

Step 6: Find the Number of Adults

Substitute \( c = 1500 \) back into the equation for \( a \):\[ a = 2200 - 1500 \]\[ a = 700 \]

Step 7: Verify the Solution

Verify the calculated numbers satisfy both equations:1. \( c + a = 1500 + 700 = 2200 \)2. \( 1.50 \times 1500 + 4.00 \times 700 = 2250 + 2800 = 5050 \)Both conditions are satisfied, confirming the solution is correct.

Key Concepts

Variables

A variable in mathematics is a symbol that stands for a number or value that can vary. In the context of solving a system of linear equations, variables are often used to represent unknown quantities that we wish to determine. Variables are crucial for constructing mathematical expressions and equations.

In the amusement park problem, we utilize two key variables:

• \( c \): This represents the number of children who entered the park.
• \( a \): This represents the number of adults who entered the park.

These variables help us to translate the word problem scenario into mathematical equations. Each variable is linked to a specific part of the problem, making it easier to form and solve equations that represent the real-world situation.

By knowing what each variable stands for, we can proceed through the calculations systematically, ensuring that each component of the solution is addressing the correct aspect of the problem.

Substitution Method

The substitution method is a popular algebraic technique for solving systems of equations. It involves solving one of the equations for one variable, and then substituting that expression into the other equation.

This approach allows you to focus initially on solving one equation, reducing complexity step by step. For our problem:

• We first solve the equation \( c + a = 2200 \) for \( a \), giving us \( a = 2200 - c \).
• Next, we substitute \( 2200 - c \) for \( a \) in the second equation, \( 1.50c + 4.00a = 5050 \).

What follows is a simplified equation in terms of one variable, \( c \). This reduction simplifies the problem significantly, enabling us to solve for \( c \) more efficiently. Once we obtain the value of \( c \), it allows us to easily determine the corresponding value of \( a \) using the equation derived earlier.

The substitution method can always be useful when one equation can be easily manipulated to express one variable in terms of the other.

Linear Equations

Linear equations are fundamental tools in algebra that represent linear relationships between variables. They are called "linear" because they graph as straight lines. Each term in a linear equation is either a constant or the product of a constant and a single variable.

In our amusement park problem, the setup gives us two linear equations:

• \( c + a = 2200 \): This equation expresses the total number of people who entered the park.
• \( 1.50c + 4.00a = 5050 \): This equation represents the total revenue collected from ticket sales.

These equations are interconnected, yet simple to interpret. They allow us to model real-world scenarios accurately and solve for unknown quantities.

Linear equations like these are valuable in many applications, from business calculations to scientific modeling. Understanding how to work with them is key to unlocking various mathematical problems and applying these concepts practically.