Problem 81 - Algebra A Combined Function

Question

A commercial jetliner hits an air pocket and drops 250 feet. After climbing 120 feet, it drops another 178 feet. What is its overall vertical change?

Step-by-Step Solution

Answer

The overall vertical change is a drop of 308 feet.

Step 1: Initial drop

The plane initially drops 250 feet. Represent this as a vertical change of -250 feet.

Step 2: Climb

The plane climbs 120 feet after the initial drop. Represent this as a vertical change of +120 feet.

Step 3: Subsequent drop

After climbing, the plane drops another 178 feet. Represent this as a vertical change of -178 feet.

Step 4: Calculate overall change

Add the changes together to find the overall vertical change: \[ (-250) + 120 + (-178) \]

Step 5: Compute result

Calculate the expression: \[ (-250) + 120 - 178 = -308 \] This means the plane's overall vertical change is a drop of 308 feet.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In our problem, these expressions represent vertical changes in the plane’s position. By treating each movement of the plane as a separate algebraic term, we can calculate the total vertical change efficiently.

Every vertical movement of the plane—be it a climb or a drop—is an example of an algebraic expression. For instance, dropping 250 feet is represented by the expression \(-250\). Similarly, climbing 120 feet is represented as \(+120\). These expressions can be combined using integer operations, which we will explore in the next section.

Algebraic expressions help simplify complex physical phenomena into mathematical terms. This allows us to use arithmetic to find solutions to real-world problems, like determining the overall vertical movement of the plane.

Integer Operations

Integer operations include the basic arithmetic operations of addition, subtraction, multiplication, and division with whole numbers. In this exercise, we used addition and subtraction to compute the plane’s total vertical change. When dealing with operations involving integers, it's crucial to consider the sign of each number.

In this exercise, drops are represented as negative integers, and climbs as positive integers. To calculate the overall change, we sum these integer values: \((-250) + 120 + (-178)\). This operation demonstrates how integers can be combined to show net changes.

• Start by adding the first two numbers: \((-250) + 120\), which gives \(-130\).
• Then, add the result with the last number: \(-130 + (-178)\), resulting in \(-308\).

This calculation shows the total vertical change. Understanding how to work with integer operations is key to solving this type of problem.

Problem Solving

Problem solving starts with understanding the situation. In our scenario, a jetliner experiences two drops and one climb during its flight. To solve it, we break down each movement into manageable steps:

• First, identify each movement: a drop, a climb, and another drop, and express them as integers.
• Next, transform the problem into a simple arithmetic operation by using algebraic expressions and integers.
• Lastly, perform the necessary calculations to find the result.

By methodically tracking each step, we can determine the jetliner’s overall vertical change effectively. The approach demonstrated here involves organizing information and using logical reasoning to arrive at the solution. It's a practical application of both mathematical understanding and strategic thinking.

Vertical Motion in Physics

Vertical motion in physics refers to the movement along the vertical axis. In the context of this problem, it involves analyzing the jetliner's vertical movements as it rises and falls. Each of these movements is affected by gravity, often resulting in a downward motion like a drop.

To determine the overall vertical motion, consider each segment of the flight path as a separate motion event. This problem captures real-life instances, such as encountering air pockets, where vertical motion can quickly change.

Physics helps us quantify these motions using math, turning them into solvable equations. The problem illustrates a simple yet effective approach to understanding physics in action by expressing vertical changes with numbers. This way, students can connect mathematical calculations with tangible events in flight dynamics.