Problem 74 - College Algebra

Question

The equation \(P=-0.5 d+100\) describes the percentage, \(P,\) of lost hikers found in search and rescue missions when members of the search team walk parallel to one another separated by a distance of \(d\) yards. If a search and rescue team finds \(70 \%\) of lost hikers, find the parallel distance of separation between members of the search party.

Step-by-Step Solution

Answer

The parallel distance of separation between members of the search party should be 60 yards.

Step 1: Identify the Given Variables

The given variables are \(P = 70\) which is the percentage of lost hikers found and the equation \(P=-0.5 d+100\). We need to find the value of \(d\).

Step 2: Plug the Given Value into Equation

Substitute the value of \(P = 70\) into the equation, resulting in \(70 = -0.5d + 100\).

Step 3: Rearrange the equation

Rearrange the equation to solve for \(d\). You can subtract \(70\) from both sides to get \(-0.5d = 70 - 100 \), which simplifies to \(-0.5d = -30\).

Step 4: Solve for \(d\)

Divide both sides of the equation by \(-0.5\) to solve for \(d\). This results in \(d = \frac{-30}{-0.5} = 60\).

Key Concepts

Problem Solving

Solving a problem often begins with understanding what is being asked and what information is provided. In this case, the problem involves determining the separation distance between members of a search team using a given mathematical equation. First, you need to identify the key variables in the problem and understand their roles:

• Percentage of lost hikers found, denoted as \(P\)
• Distance between search team members, denoted as \(d\)

By knowing that the search team finds 70% of lost hikers and using the equation \(P=-0.5 d+100\), we have a basis for finding the distance \(d\). This understanding forms the backbone of how you will approach solving the problem.

Algebraic Manipulation

Algebraic manipulation is a powerful tool for solving equations. With our problem, the goal is to isolate the variable \(d\), which represents the distance between searchers.First, substitute the given value of \(P\) into the equation:\[ 70 = -0.5d + 100 \]The next step involves making \(d\) the subject of the formula. Start by rearranging the terms:

• Subtract 100 from both sides to maintain equality: \(70 - 100 = -0.5d\)
• The equation simplifies to \(-30 = -0.5d\)

Finally, solve for \(d\) by dividing both sides of the equation by \(-0.5\):\[ d = \frac{-30}{-0.5} = 60 \]Through these steps, we used algebraic manipulation to uncover that each searcher should be spaced 60 yards apart.

Mathematical Modeling

Mathematical modeling is about representing real-world situations using mathematical concepts and equations. The equation \(P = -0.5 d + 100\) models the relationship between the distance \(d\) between searchers and the percentage \(P\) of lost hikers found.Here's how it works:

• The coefficient \(-0.5\) indicates how substantially the percentage changes with each unit increase in distance.
• The constant \(100\) represents the ideal scenario where searchers are close, maximizing the percentage of hikers found.

By applying mathematical modeling, the search and rescue team can optimize their efforts by understanding and adjusting the spacing based on this predictive model. This highlights a practical use of math in real life, offering solutions to improve efficiency and effectiveness.