Problem 40 - Elementary Algebra Within Reach

Question

Distance Two planes leave Orlando International Airport approximately the same time and fly in opposite directions (see figure). Their speeds are 510 miles per hour and 600 miles per hour. How far apart will the planes be after \(1 \frac{1}{2}\) hours?

Step-by-Step Solution

Answer

The two planes will be 1665 miles apart after 1 and a half hours.

Step 1: Compute distance travelled by the first plane

The distance travelled by the first plane is computed by multiplying the speed of the plane, which is 510 miles per hour, by the time. Taking into account that the time is 1.5 hours (which we derive from \(1\frac{1}{2}\) hours), the calculation will be \(1.5 \times 510 = 765\) miles.

Step 2: Compute distance travelled by the second plane

The distance travelled by the second plane is computed similarly to step 1. This time, the speed is 600 miles per hour, so the calculation will be \(1.5 \times 600 = 900\) miles.

Step 3: Compute total distance between the planes

The total distance between the planes is given by the sum of the distances travelled by each plane. It can be calculated as \(765 + 900 = 1665\) miles.

Key Concepts

Speed and Distance

When calculating distance, the fundamental formula to use is \( \text{Distance} = \text{Speed} \times \text{Time} \). This formula allows you to find out how far an object has traveled based on how fast it was going and for how long.
For instance:

• If a plane is flying at a speed of 510 miles per hour and we want to know the distance it covers in 1.5 hours, we multiply: \( 510 \times 1.5 = 765 \) miles.
• Similarly, for a plane traveling at 600 miles per hour for 1.5 hours, the distance is \( 600 \times 1.5 = 900 \) miles.

It's important to ensure that the time is in the correct units (hours in this case) when using this formula. This type of calculation is useful for determining distances in practical scenarios like travel and transport.

Opposite Directions

The concept of objects moving in opposite directions means they are moving away from each other, each on its own path. In such cases, the total distance between the two objects is the sum of the distances each has traveled.
Applying this thought process here:

• First plane travels 765 miles eastward.
• Second plane travels 900 miles westward.

To find how far apart the two planes will be, we add the distance traveled by each: \( 765 + 900 = 1665 \) miles.
Understanding directionality is crucial in problems like these, as it impacts how distances should be combined, and it helps hypothesize the overall scenario in real life.

Algebraic Calculations

Algebra can simplify many distance-related problems, helping you solve for unknowns or verify calculations. Here, arithmetic operations involve straightforward multiplication and addition, typical for systems where determining separation is needed.
Let's break it down:

• Multiply the speed of each plane by the time: Plane 1 is \( 510 \times 1.5 \), and Plane 2 is \( 600 \times 1.5 \).
• Add the results to find the total separation: The arithmetic sum \( 765 + 900 = 1665 \) confirms the total distance apart.

Algebra extends beyond simple operations if complexities arise, like finding time when distances and speeds are known. Mastery of algebraic principles strengthens the ability to handle a variety of scenarios efficiently.