Problem 32 - Algebra A Combined Function

Question

The most popular amusement park in the world (according to annual attendance) is Tokyo Disneyland, whose yearly attendance in thousands can be approximated by the equation \(y=1201 x+16,507\) where \(x\) is the number of years after 2000 . In second place is Walt Disney World's Magic Kingdom, whose yearly attendance, in thousands, can be approximated by \(y=-616 x+15,400\). Find the last year when attendance in Magic Kingdom was greater than attendance in Tokyo Disneyland. (Source: Amusement Business)

Step-by-Step Solution

Answer

The last year was 2000.

Step 1: Identify the Equations

We have two equations representing attendance, where:- Tokyo Disneyland: \(y = 1201x + 16,507\)- Magic Kingdom: \(y = -616x + 15,400\). We need to find the year \(x\) when Magic Kingdom's attendance is greater than Tokyo Disneyland's.

Step 2: Set Up the Inequality

Use the given equations to set up the inequality to solve for when Magic Kingdom's attendance exceeds that of Tokyo Disneyland:\[-616x + 15,400 > 1201x + 16,507\].

Step 3: Solve the Inequality for \(x\)

Rearrange the inequality to isolate \(x\):- Subtract \(1201x\) from both sides: \[-616x - 1201x + 15,400 > 16,507\]- Combine like terms: \[-1817x + 15,400 > 16,507\]- Subtract 15,400 from both sides: \[-1817x > 1,107\]- Divide by -1817, and don't forget to flip the inequality sign: \[x < -0.609\].

Step 4: Interpret the Result

Since \(x\) represents the number of years after 2000, \(x < -0.609\) implies the last full year when attendance in Magic Kingdom was greater than Tokyo Disneyland was 2000 (as \(x\) should be a whole number).

Key Concepts

Solving Inequalities

When solving inequalities, we are looking to find the values for a variable that satisfy the inequality. This process is similar to solving equations, but there are key differences to remember. Inequalities come in four types: greater than (<), less than (>), greater than or equal (\geq), and less than or equal (\leq). It's important to get the variable on one side of the inequality sign alone.
\[\text{Here are steps to follow when solving inequalities:}\]

• Isolate the Variable: Rearrange the inequality so that all terms with the variable are on one side and constant terms are on the other.
• Combine Like Terms: Perform any addition or subtraction needed to simplify each side of the inequality.
• Divide or Multiply: If you divide or multiply both sides by a negative number, flip the inequality sign.
• Solve for the Variable: Perform any final arithmetic needed to solve for the variable's value.

Practice helps in recognizing the difference between equations and inequalities and when to flip the inequality sign.

Linear Equations

Linear equations are the backbone for understanding many algebraic concepts. A linear equation is any equation that can be written in the form \(ax + b = c\). It represents a straight line when graphed on a coordinate plane and shows a constant relationship between two quantities.
Linear equations appear in many formats, such as:

• Standard form: \(Ax + By = C\)
• Slope-intercept form: \(y = mx + b\), with \(m\) being the slope and \(b\) the y-intercept
• Point-slope form: \(y - y_1 = m(x - x_1)\)

Understanding how to manipulate different forms helps in solving various algebraic problems quickly and accurately. When dealing with word problems, converting the scenario into a linear equation is often the first step.

Word Problems in Algebra

Word problems in algebra require us to translate a real-world situation into a mathematical model. This involves identifying knowns and unknowns, setting up equations or inequalities, and solving them to find the desired values. The key is breaking down the problem into manageable parts.
Here's how to approach word problems:

• Read Carefully: Understand the problem and what is being asked. Look for keywords that indicate mathematical operations.
• Define Variables: Assign variables to unknown quantities and express known quantities in terms of these variables.
• Set Up Equations or Inequalities: From the problem details, form equations or inequalities that model the situation.
• Solve and Interpret: Solve the equations or inequalities and make sure to put your answer in context of the problem.

Having a systematic approach can reduce confusion and errors, ensuring an effective solution.

Educational Mathematics

Educational mathematics aims to develop strong analytical and problem-solving skills. By engaging with mathematical concepts through exercises, learners develop critical thinking abilities that are applicable in many areas of life.
Key aspects include:

• Conceptual Understanding: Grasping the 'why' behind techniques rather than just memorizing steps.
• Real-World Applications: Connecting mathematics with daily life through practical examples.
• Continuous Practice: Enhancing proficiency and confidence in mathematical abilities through regular and varied practice.
• Teaching Strategies: Using diverse teaching methods to adapt to different learning styles and abilities.

Educational mathematics not only prepares students for exams but also nurtures an appreciation for how math structures our understanding of the world.