Problem 122 - Algebra and Trigonometry

Question

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 \(2x + y \leq 8000\).

Step 1: Identify Given Information

The weight of each container of bottled water is 20 pounds, and this is represented by the variable \(x\). The weight of each medical kit is 10 pounds, represented by the variable \(y\). Every plane has a weight limit of 80,000 pounds.

Step 2: Formulate the Inequality

The sum of the weights of the water bottles and medical kits cannot exceed the total weight limit of the plane. This relationship can be expressed as an inequality: \(20x + 10y \leq 80000\). This equation states that the total weight of the containers (20 times the number of containers) plus the total weight of the kits (10 times the number of kits) must be less than or equal to the plane's weight limit.

Step 3: Simplifying the Inequality

To make the inequality simpler, we can divide each term in the inequality by 10. The simplified inequality would then be \(2x + y \leq 8000\)

Key Concepts

Weight Restrictions

Weight restrictions are essential when considering the safe operation and capacity of vehicles or machines, such as airplanes. In our given problem, the plane can carry a maximum of 80,000 pounds, which includes both bottled water and medical supplies. This restriction serves as a maximum threshold you cannot exceed. Understanding this concept is crucial:

• It ensures safety by preventing overloading, which could lead to potential risks or accidents.
• It helps in resource allocation, noting how much of each item can fit within allowable limits.

Recognizing weight limits is about managing resources effectively. Breaking down these limits into manageable parts helps in planning what needs to be shipped versus what can safely remain. Hence, always consider individual weights (20 pounds per water container and 10 pounds per medical kit) against the overall limit.

Inequality Modeling

Inequality modeling is a mathematical way to show limitations and constraints. In our case, it's about the relationship between the number of water bottles (\(x\)) and the number of medical kits (\(y\)) that satisfy the plane's capacity.To create the inequality, we understand that:

• The weight of each water container is added along with each medical kit's weight.
• These combined weights must not exceed 80,000 pounds, the plane's capacity.

This idea is expressed by the inequality\(20x + 10y \leq 80000\).Inequality modeling allows us to represent real-world scenarios with mathematical expressions. We capture restrictions by separating and understanding each component's contribution to the total weight.Ultimately, inequality modeling serves as a guide to maintaining balance and ensuring conditions are satisfied without exceeding limits.

Simplifying Inequalities

Simplifying inequalities helps make mathematical expressions easier to work with by reducing their complexity. Simplification is crucial for:

• Facilitating easier interpretation and problem-solving.
• Making calculations more straightforward and reducing potential errors.

In this context, the original inequality\(20x + 10y \leq 80000\)can be simplified by dividing each term by 10. This gives us\(2x + y \leq 8000\).By reducing the coefficients, we simplify the expression without changing the inequality's meaning. It provides a clearer picture of how many water containers and medical kits can fit by simplifying the calculation needed for logistical planning.Simplification doesn't alter the situation described; instead, it makes interpreting the data in practical terms far easier. Understanding how to simplify ensures that students can more easily apply these concepts in various real-world scenarios.