Problem 118 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane. The bottled water weighs 20 pounds per container and medical kits weigh 10 pounds per kit. Each plane can carry no more than \(80,000\) pounds. If \(x\) represents the number of bottles of water to be shipped per plane and \(y\) represents the number of medical kits per plane, write an inequality that models each plane's \(80,000\) -pound weight restriction.

Step-by-Step Solution

Answer

The inequality that models each plane's 80,000-pound weight restriction is \(20x + 10y ≤ 80000\).

Step 1: Establish the Variables

Let \(x\) represent the number of containers of bottled water, each weighing 20 pounds. Let \(y\) represent the number of medical kits, each weighing 10 pounds.

Step 2: Setup the Linear Inequation Representing the Weight Limitation

The combined weight of the water containers and the medical kits cannot exceed 80,000 pounds. In mathematical terms, this can be expressed as \(20x + 10y ≤ 80000\). This inequality describes the situation; the left side represents the total weight of the cargo, and the right side represents the maximum weight the plane can carry.

Key Concepts

Variables

In the world of mathematics, variables are used to represent quantities that can change or vary. In this exercise, we have two variables, which are:

• \( x \) for the number of containers of bottled water.
• \( y \) for the number of medical kits.

The purpose of using these variables is to create a mathematical model that represents a real-world situation.
Variables are placeholders. They let us work with unknown values and explore their relationships within an equation.

Weight Limitation

A weight limitation is a constraint or a condition that specifies how much load something can carry. In this exercise, it relates to the plane's ability to carry a certain total weight.
Each plane cannot carry more than 80,000 pounds, which is critical for safety and efficiency.
This weight limitation helps define the equation. It informs the boundary conditions the mathematical model must adhere to, ensuring solutions are realistic and applicable in the real world.

Linear Equation

A linear equation is a mathematical statement that describes a line in a coordinate plane. The equation in this context uses the weights of water containers and medical kits.
The equation derived is:

• \( 20x + 10y \leq 80000 \)

This inequality indicates that the total weight contribution of the containers (\( 20x \)) and medical kits (\( 10y \)) should not exceed 80,000 pounds.
This equation is linear because both variables, \( x \) and \( y \), are raised to the power of one.
It visually represents a boundary line on a graph, splitting the possible weight combinations that are permissible from those that exceed the specified limit.

Maximum Weight

The concept of maximum weight refers to the largest amount that can be carried without compromising safety or structural integrity.
For the plane, this maximum permissible load is set at 80,000 pounds.
By establishing a maximum weight, we can make decisions about how many containers of water and medical kits can be safely loaded onto a plane.

• Effectively distributing this weight requires solving the inequality \( 20x + 10y \leq 80000 \).

Solutions should account for a practical arrangement of cargo while adhering to the upper limits, ensuring safe transportation of goods.