Problem 122 - Beginning and Intermediate Algebra

Question

Solve each problem. A pilot announces to the passengers that the current altitude of their plane is \(34,000 \mathrm{ft}\). Because of turbulence, the pilot is forced to descend \(2100 \mathrm{ft}\). Write the new altitude as a signed number.

Step-by-Step Solution

Answer

The new altitude is +31,900 feet.

Step 1: Understand the Problem

The plane is initially at an altitude of 34,000 feet. Due to turbulence, the pilot descends 2,100 feet. The goal is to determine the new altitude as a signed number.

Step 2: Set Up the Equation

Represent the initial altitude and the change in altitude in an equation. The initial altitude is 34,000 feet and the plane descends, which is represented as a negative change: Initial altitude: 34,000 feet Descend: 2,100 feet Equation: 34,000 - 2,100

Step 3: Perform the Calculation

Subtract 2,100 feet from the initial altitude of 34,000 feet: 34,000 - 2,100 = 31,900

Step 4: Write the Answer as a Signed Number

After descending, the new altitude of the plane is 31,900 feet. Since the plane is descending, the change is represented as a negative: New altitude as a signed number: +31,900 feet

Key Concepts

Subtraction

Subtraction is a fundamental operation in mathematics that tells us how much one quantity is less than another. In our exercise, we subtracted the descent in altitude from the initial altitude to find the new altitude.

The equation was: \[ 34,000 - 2,100 \]

By performing this operation, we find that the new altitude is 31,900 feet.

Subtraction helps in various real-world problems. Whether you're calculating change, determining differences, or finding how much of something is left, subtraction is essential.

It's important to place the larger number first when setting up a subtraction equation, especially when dealing with positive and negative quantities. Doing so ensures clarity and accuracy.

Signed Numbers

Signed numbers include both positive and negative numbers and help represent changes in quantity, direction, or position. For example, going up can be positive, whereas going down in altitude is negative.

In our exercise, the descent was a negative change:

\[ \text{Descend:} -2,100 \]

When we subtracted, the negative change decreased the initial altitude:

\[ 34,000 - 2,100 = 31,900 \]

Understanding signed numbers helps solve problems involving gains or losses, temperature changes, and even financial calculations.

Remember, when subtracting a negative number, it effectively adds the absolute value of that number. For instance, \[ 10 - (-5) = 15 \] because subtracting a negative is like adding a positive.

Real-World Applications

Math isn't just abstract; it's deeply connected to our daily lives. In our exercise, the altitude problem represents a real-life situation that pilots and aviation professionals face.

Here are a few real-world applications of solving equations:

• **Aviation:** Pilots use math to calculate fuel consumption, distances, and altitudes.
• **Finance:** Banks and individuals use it for interest calculations, mortgages, and budgeting.
• **Engineering:** Engineers use equations to design structures, develop software, and solve construction issues.
• **Healthcare:** Doctors and pharmacists calculate dosages, medical statistics, and treatment plans.
• **Everyday Life:** From baking recipes to planning trips, we frequently use math to make accurate decisions.

By practicing solving these equations, students can better prepare themselves for these real-world challenges.