Problem 44 - Intermediate Algebra

Question

Two marathon runners leave the starting gate, one running 12 mph and the other 10 mph. If they maintain the pace, how long will it take for them to be one- quarter of a mile apart?

Step-by-Step Solution

Answer

They will be 0.25 miles apart in 7.5 minutes.

Step 1: Understand the problem

We have two runners starting at the same time. One is running at 12 mph, and the other at 10 mph. We are asked to find the time it takes for them to be 0.25 miles apart.

Step 2: Set up the equation

Since they are moving in the same direction, the distance between them after time \( t \) can be described by the equation \( d = (12t - 10t) = 2t \). We need this distance \( d \) to be 0.25 miles.

Step 3: Solve for time \( t \)

Set up the equation based on Step 2: \( 2t = 0.25 \). Solve for \( t \) by dividing both sides of the equation by 2.

Step 4: Calculate \( t \)

From the equation \( 2t = 0.25 \), divide both sides by 2 to get \( t = 0.125 \) hours.

Step 5: Convert time into minutes

Since time is often more meaningful in minutes, convert 0.125 hours to minutes by multiplying by 60. \( t = 0.125 \times 60 = 7.5 \) minutes.

Key Concepts

Rate of Change

When we delve into rate of change, we often reference the speed at which something occurs over time. In the context of our marathon runners, the rate of change refers to their speeds. For runner A, this is 12 mph, and for runner B, it's 10 mph. The rate here is crucial in determining how quickly the distance between the two runners increases. This is because a higher rate (or faster speed) results in a quicker change over a specific period. In problems like this, understanding rate of change helps us determine how variables like distance and time relate. The time it takes for two objects to be a certain distance apart relies on how fast they are moving relative to each other. Here, we deduced that the rate at which the distance between the runners is growing is 2 mph, since it is the difference between their speeds (12 mph - 10 mph). Therefore, the runners, moving together initially, gradually increase the gap by 2 miles every hour.

Equations in Algebra

Algebraic equations help us find unknown variables in a systematic manner by setting up mathematical sentences. In our example, we needed to understand when the runners reach a separation of 0.25 miles. To find this, we considered how fast the separation increases over time.The relationship between distance, rate, and time is often expressed as the equation: \[ d = rt \]Where \( d \) is the distance, \( r \) is the rate of change (in this case the difference in the runners' speeds), and \( t \) is time. In setting up the equation, we have:

• \( d = 0.25 \) miles
• \( r = 2 \) mph

Thus, our equation becomes \( 2t = 0.25 \). Solving for \( t \) involves isolating this variable, which makes it clear we're looking for when their difference reaches 0.25 miles. A simple division of both sides by 2 yields the solution for \( t \).

Unit Conversion

Unit conversion is a key skill, allowing us to present results in a more interpretable way. In our case, after solving the equation \( 2t = 0.25 \), we found that \( t = 0.125 \) hours. However, this isn't typically how people think about time. Converting hours into minutes makes the outcome clearer to most.Here's how conversion works:

• There are 60 minutes in one hour.
• To convert hours to minutes, multiply by 60.

For our problem, converting 0.125 hours into minutes concludes with:\[ 0.125 \times 60 = 7.5 \text{ minutes} \]This final conversion helps create clarity, making it easier to visualize and comprehend the time it takes for the runners to be 8 mile apart.