Problem 20 - College Algebra

Question

Ben starts walking along a path at \(4 \mathrm{mi} / \mathrm{h} .\) One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at \(6 \mathrm{mi} / \mathrm{h}\). How long will it be before Amanda catches up to Ben?

Step-by-Step Solution

Answer

Amanda will catch Ben in 3 hours.

Step 1: Define Variables and Setup Equations

Let the time taken by Amanda to catch Ben be \( t \) hours. Since Amanda starts 1.5 hours after Ben, the time Ben has been walking is \( t + 1.5 \) hours. The distance Ben and Amanda travel will be equal when Amanda catches Ben. We setup the equation for distance:\[4(t + 1.5) = 6t\]

Step 2: Expand and Simplify Equation

Expand the equation:\[4t + 6 = 6t\]Subtract \(4t\) from both sides to simplify:\[6 = 6t - 4t\]

Step 3: Solve for Time \( t \)

Simplify \(2t = 6\) by dividing both sides by 2:\[t = 3\]Thus, it takes Amanda 3 hours to catch up to Ben.

Key Concepts

Linear Equations

Linear equations are at the heart of many mathematical problems, and they enable us to solve for unknown values. A linear equation is essentially an equation involving only linear terms, which means each term is either a constant or the product of a constant and a single variable. When graphed, a linear equation will create a straight line.

In the problem at hand, we set up a linear equation based on the distances covered by Ben and Amanda. Both distances need to be equal for Amanda to catch Ben, so we equate them. Ben's distance is expressed as \(4(t + 1.5)\), where he walks at 4 miles per hour for \(t + 1.5\) hours. Amanda's distance is represented by \(6t\), as she jogs at 6 miles per hour for \(t\) hours. Setting these equations equal allows us to solve for \(t\), Amanda’s time in motion, ultimately determined as 3 hours in this scenario.

Solving Word Problems

Solving word problems can often be challenging, but breaking them down into smaller, manageable parts can simplify the process. Here's a helpful approach:

• Read the problem carefully and identify the main facts and question.
• Decide what you need to find and choose clear variables to represent unknowns.
• Translate the word problem into mathematical expressions or equations.
• Use logical reasoning and known math concepts to solve the equations.

In this particular problem, we identified that the main goal was to determine when Amanda catches up to Ben. We chose \(t\) to represent the time after Amanda starts jogging. From here, leveraging the relationship between distance, speed, and time allowed us to formulate and solve a linear equation. This systematic breakup makes complex word problems much easier to tackle.

Distance Formula

The distance formula is a foundational concept in solving motion problems. It relates distance, speed, and time with the simple equation: \(\text{Distance} = \text{Speed} \times \text{Time}\). This formula is particularly useful for determining how far someone travels over a period.

In our scenario, both Ben and Amanda travel along the same path at different speeds. For Ben, distance traveled is given by his speed \(4 \text{ miles per hour} \) multiplied by his time \( (t + 1.5) \). For Amanda, it is her speed \(6 \text{ miles per hour} \) times her time \(t\). Applying the distance formula and setting these distances equal allows us to determine the time it takes for Amanda to catch up, demonstrating the powerful utility of the distance formula in real-world contexts.