Problem 53 - Algebra for College Students

Question

Solve each of Problems \(47-62\) by setting up. Two airplanes leave Chicago at the same time and fly in opposite directions. If one travels at 450 miles per hour and the other at 550 miles per hour, how long will it take for them to be 4000 miles apart?

Step-by-Step Solution

Answer

It will take 4 hours for the airplanes to be 4000 miles apart.

Step 1: Understand the Problem

Two airplanes are leaving from the same point and traveling in opposite directions. One airplane is traveling at 450 miles per hour and the other at 550 miles per hour. We need to find out how long it will take for the total distance between them to be 4000 miles.

Step 2: Determine the Equation

To find the time it takes for the airplanes to be 4000 miles apart, we need to set up the equation based on the idea that distance equals speed times time. If \( t \) is the time in hours they have been traveling, then the distance traveled by the first airplane is\( 450t \) and by the second airplane is \( 550t \). The total distance apart is \( 450t + 550t = 4000 \).

Step 3: Simplify the Equation

Combine the like terms in the equation \( 450t + 550t \). This gives us \( 1000t = 4000 \).

Step 4: Solve for Time \( t \)

To solve for \( t \), divide both sides of the equation by 1000: \[ t = \frac{4000}{1000} \] Calculate the result to find \( t \).

Step 5: Calculate the Result

Perform the division: \[ t = 4 \] Therefore, it will take 4 hours for the airplanes to be 4000 miles apart.

Key Concepts

Rate of Travel

When thinking about the rate of travel, it's essential to understand that it represents how fast something moves over a certain period. In most cases, this is expressed as a distance covered per unit of time, such as miles per hour or kilometers per hour.
For the airplane problem, one plane travels at a speed of 450 miles per hour, and the other at 550 miles per hour. This indicates that in one hour, the first airplane will cover 450 miles and the second will cover 550 miles.

Knowing the rate of travel helps us compare how fast different objects move and determine how long it will take to travel a certain distance. Always keep in mind:

• The speed is constant for each airplane, meaning they maintain a steady pace.
• Different speeds mean different distances will be covered over the same time period.

This understanding is vital in setting up word problems involving distance, especially when multiple objects are in motion.

Solving Equations

To resolve distance problems, like determining how long it takes for two airplanes to be a certain distance apart, setting up and solving equations is crucial.
In this scenario, we use the formula: distance equals rate times time. For each airplane, this becomes:

• Distance for the first airplane: \(450t\)
• Distance for the second airplane: \(550t\)

These two equations sum up to the total distance:450t + 550t = 4000, reflecting the combined distance they need to be apart.

We can simplify equations by combining like terms, which here gives us 1000t = 4000. To find time \( t \), solve the equation by dividing both sides by 1000, resulting in \( t = 4 \).
This simple arithmetic gives us the time as 4 hours. The step-by-step simplification is crucial here, as it paves the way for solving straightforward distance problems.

Word Problems

Solving word problems requires a clear understanding of the problem's context and details. In this particular airplane scenario, we navigate through the problem by:

• Identifying the given information: Two airplanes leave a point traveling in opposite directions, each with a known speed.
• Establishing what you need to find: How long it takes for them to be 4000 miles apart.

Word problems might look complicated, but breaking them down into small steps helps. Extracting information to form equations based on known relationships, like distance and rate, is a powerful strategy.

Word problems often require skills such as:

• Interpreting verbal information into mathematical equations or expressions.
• Decoding what you are solving for, in this case, the time.

Approaching word problems methodically, and practicing these interpretation skills, will make navigating through them much easier, turning them into simpler, approachable challenges.