Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Two children, who are 224 meters apart, start walking toward each other at the same instant at rates of \(1.5 \mathrm{~m} / \mathrm{sec}\) and \(2 \mathrm{~m} / \mathrm{sec}\), respectively (see the figure). (a) When will they meet? (b) How far will each have walked?

Step-by-Step Solution

Answer

(a) 64 seconds; (b) 96 meters (first child), 128 meters (second child).

Step 1: Define Variables and Constants

Let's assign variables for the distances each child covers to make it easier to work with. Let \( x \) be the distance covered by the first child, who walks at a speed of \( 1.5 \, \text{m/s} \) and \( y \) be the distance covered by the second child, who walks at a speed of \( 2 \, \text{m/s} \). Initially, they are 224 meters apart. We need to find when they meet and how far each child has walked.

Step 2: Set Up Distance Meeting Equation

Since they start 224 meters apart, the combined distances they cover when they meet is 224 meters. Therefore, the equation is:\[x + y = 224\]Moreover, since distance equals speed times time, we can express \( x \) and \( y \) in terms of time \( t \):\[x = 1.5t, \quad y = 2t\]Substitute these into the equation:\[1.5t + 2t = 224\]

Step 3: Solve for Time \(t\)

Combine the terms in the equation:\[3.5t = 224\]Now solve for \( t \) by dividing both sides by 3.5:\[t = \frac{224}{3.5} = 64 \, \text{seconds}\]

Step 4: Calculate Distances for Each Child

Now that we have the time \( t = 64 \) seconds, we can find how far each child has walked.For the first child:\[x = 1.5t = 1.5 \times 64 = 96 \, \text{meters}\]For the second child:\[y = 2t = 2 \times 64 = 128 \, \text{meters}\]

Step 5: Verify the Solution

The total distance walked by both children should sum to 224 meters, verifying that we have the correct individual distances:\[96 + 128 = 224\]This confirms that the calculations are correct and consistent with the problem's initial condition.

Key Concepts

Distance Formula

In algebra, the distance formula is crucial for solving problems where you need to find how far entities have traveled. In this problem, the two children are 224 meters apart and the challenge is to find the point in time and space where they meet.

The formula essentially helps us equate the total distance traveled by both parties. Here, this is given by:

• Distance = Rate × Time

Each child walks towards each other, covering a part of the total 224 meters.
Their combined distance as they meet is represented as:

• \[ x + y = 224 \]

Using this equation, we effectively apply the distance formula to set up and solve the problem.

Simultaneous Equations

Simultaneous equations come into play when we have two variables and we need to find a solution that satisfies both equations. In our exercise, these equations will be based on the children’s distances.

We know:

• First child's distance: \( x = 1.5t \)
• Second child's distance: \( y = 2t \)
• Total Distance: \( x + y = 224 \)

By substituting the distance expressions into the main equation, we form a single equation:

• \[ 1.5t + 2t = 224 \]

Now, instead of solving for two distinct variables, we solve for just one — the time \( t \). This process of substitution helps us find the time when both children will meet.

Rate-Time-Distance Problems

Rate-Time-Distance problems are a staple in word problems. They involve finding one of the three variables — rate, time, or distance — when the other two are known. Here, the children walking towards each other present one such problem.

You typically start by identifying:

• Rate: Speed of the children, \( 1.5 \text{ m/s} \) and \( 2 \text{ m/s} \)
• Time: The time \( t \) they spend walking
• Distance: The spatial distance covered by them, adding up to \( 224 \text{ m} \)

Such problems boil down to basic algebraic equations where you apply the core formula:

• Distance = Rate × Time

By mastering how to rearrange and solve these equations, tackling word problems becomes much simpler.

Word Problems in Algebra

Word problems can seem daunting as they require translation of text into mathematical equations. The key to mastering them is identifying variables and forming appropriate equations.

This problem, at its heart, is about real-world movement modeled by algebraic expressions. Start by:

• Identifying what's being asked: When will they meet? How far has each walked?
• Breaking down the information given: Their rates and the distance initially between them
• Forming equations to solve for unknowns

The success with word problems increases as you practice translating scenario descriptions into variables and equations, leading to logical solutions.