Problem 21 - Algebra for College Students

Question

Two steamers leave a port on a river at the same time, traveling in opposite directions. Each is traveling \(22 \mathrm{mph}\). How long will it take for them to be \(110 \mathrm{mi}\) apart?

Step-by-Step Solution

Answer

It will take 2.5 hours.

Step 1 - Determine the combined speed

The two steamers are traveling in opposite directions, so their speeds add together. Each steamer is traveling at 22 mph. Therefore, their combined speed is: \[ 22 \text{ mph} + 22 \text{ mph} = 44 \text{ mph} \]

Step 2 - Use the distance formula

The formula to find time when distance and speed are known is given by: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]We know the combined speed is 44 mph and the desired distance apart is 110 miles.

Step 3 - Calculate the time

Insert the known values into the formula from Step 2: \[ \text{Time} = \frac{110 \text{ miles}}{44 \text{ mph}} \]This simplifies to: \[ \text{Time} = 2.5 \text{ hours} \]

Key Concepts

relative speed

When objects move in relation to each other, we can talk about their **relative speed**.
In this exercise, the two steamers travel in opposite directions, meaning we need to add their speeds to find out how fast they move apart.
Each steamer travels at 22 mph, so:
\[ 22 \text{ mph} + 22 \text{ mph} = 44 \text{ mph} \]
This combined speed is their **relative speed**.
Using relative speed simplifies complex problems and aids in understanding how distances change over time.

travel time calculation

To find out how long it will take the steamers to be a certain distance apart, we use the travel time formula.
The formula is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
We already know that the steamers' combined speed is 44 mph. The problem states they need to be 110 miles apart.
Plugging in these values:
\[ \text{Time} = \frac{110 \text{ miles}}{44 \text{ mph}} \] = 2.5 hours
This formula helps us calculate the time needed to cover a specific distance at a known speed.

distance-time relationship

The relationship between distance, time, and speed is fundamental in understanding motion.
The distance formula, \[ \text{Distance} = \text{Speed} \times \text{Time} \] can be rearranged to solve for time or speed.
The three components are interconnected:

• Distance tells us how far an object has traveled.
  
• Speed indicates how fast it's traveling.
  
• Time reveals how long it's been traveling.
  

In our problem, we used:

• Distance = 110 miles
  
• Speed = 44 mph (combined)
  
• Time = 2.5 hours
  

Understanding this relationship aids in solving many real-world problems involving motion.