Problem 293 - Elementary Algebra

Question

In the following exercises, solve. Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?

Step-by-Step Solution

Answer

3 hours

Step 1: Determine the Total Speed

Since Cindy and Richard are traveling in opposite directions, add their speeds together. Cindy's speed is 18 miles per hour, and Richard's speed is 14 miles per hour. Therefore, their combined speed is: \[ 18 + 14 = 32 \text{ miles per hour} \]

Step 2: Write the Distance Formula

The formula for distance is: \( \text{Distance} = \text{Speed} \times \text{Time} \)Rearrange the formula to solve for time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

Step 3: Substitute the Values

Substitute the values into the rearranged distance formula. The distance is 96 miles, and the combined speed is 32 miles per hour:\[ \text{Time} = \frac{96 \text{ miles}}{32 \text{ miles per hour}} \]

Step 4: Calculate the Time

Perform the division to find the time:\[ \text{Time} = \frac{96}{32} = 3 \text{ hours} \]

Key Concepts

Relative Speed

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. Cindy rides north at 18 mph, and Richard rides south at 14 mph. Their combined speed is \( 18 + 14 = 32 \text{ miles per hour} \). If you think about it, it's like one of them is moving at 32 mph relative to the other. This concept is useful when you want to find out how quickly the distance between the two increases.

Distance Formula

To calculate distance, use the formula: \( \text{Distance} = \text{Speed} \times \text{Time} \). Here, 'Speed' is how fast something is moving, and 'Time' is for how long it moves. If you know the distance and speed, you can rearrange the formula to find time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). For example, if Cindy and Richard are 96 miles apart and their combined speed is 32 mph, you can find the time it takes by dividing the distance by the speed.

Time Calculation

Once you rearrange the distance formula to solve for time, it becomes \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). In our problem, plugging in 96 miles for Distance and 32 mph for Speed, we get: \( \text{Time} = \frac{96}{32} \). Performing the division, the time comes out to be 3 hours. This means Cindy and Richard will be 96 miles apart after 3 hours of riding in opposite directions.

Opposite Directions

When two objects move in opposite directions, the distance between them increases faster compared to when either of them is stationary or moving in the same direction. The speeds add up, making the relative speed high. For instance, Cindy going north at 18 mph and Richard going south at 14 mph makes the distance between them grow at 32 mph. Understanding this helps to quickly calculate time and distance when objects move away from each other.