Question

Travel Time Two cars start at the same time at a given point and travel in the same direction at constant speeds of 40 miles per hour and 55 miles per hour. After how long are the cars 5 miles apart?

Short answer

The cars will be 5 miles apart after 1/3 hour, which is equivalent to 20 minutes.

Step-by-step solution

Identify Given Information

The cars are moving in the same direction with one car moving at 40 miles per hour and the other at 55 miles per hour.

Calculate Relative Speed

The cars are moving in the same direction, therefore the relative speed will be the difference of the two speeds. This is calculated as follows: Relative speed = Speed of car B - Speed of car A = 55 miles/hr - 40 miles/hr = 15 miles/hr.

Calculate the Time

The question asks for how long are the cars 5 miles apart, to find this we use the formula: Time = Distance / Speed. Here, the distance is 5 miles while the speed is the relative speed we found earlier. Hence, Time = 5 miles / 15 miles/hr = 1/3 hr

Key concepts

Algebraic Problem-Solving

Understanding algebraic problem-solving is crucial when tackling questions about movement and distances. It involves identifying variables, forming equations, and solving for unknowns.

Let's take our cars’ problem. The first step is always about understanding the given information; in our case, the speeds of two cars and the desired separation distance. After identifying these, we frame our unknown variable, which is the time needed for the cars to be 5 miles apart. Using algebra, we set up an equation representing the scenario: \( \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \). In the initial steps of solving, we calculated the relative speed. Now we simply plug that, along with the 5-mile distance, into our equation to solve for time.

When crafting algebraic solutions, always keep in mind to isolate and solve for the unknown variable using basic operations—addition, subtraction, multiplication, and division. This lays a strong foundation for the problem-solving process.

Constant Speed

The concept of constant speed is vital in many physics and mathematics problems involving motion. A constant speed implies that an object covers equal distances in equal intervals of time, which simplifies many calculations.

Referring back to our problem, both cars travel at constant speeds – one at 40 mph and the other at 55 mph. Understanding that their speeds don't change allows us to predict their positions at any given time. When two objects move at constant speeds, especially in the same direction as our cars, we can calculate their relative speed simply by taking the difference between their speeds. This concept is fundamental because it gives us a single value with which to work when determining the time it takes for one object to gain a specific distance over the other.

Remember, in real-world situations speeds may not remain constant due to various factors such as traffic or road conditions, but for the purpose of these exercises, we assume constant speeds to create clear mathematical models.

Distance Formula

In problems that involve movement and measuring gaps between moving objects, the distance formula becomes an indispensable tool. The formula \( \text{Distance} = \text{Speed} \times \text{Time} \) helps in calculating one of the three values when the other two are known.

In our exercise, we are dealing with a modified use of the formula to find time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). This is derived from rearranging the basic distance formula. Given a specific distance you want to be covered and the speed at which you're moving towards or away from something, one can calculate the time it takes to cover that distance.

A clear comprehension of how to manipulate the distance formula allows for solving a wide range of problems, not just with constant speeds, but also with varying rates, as long as these rates can be averaged over the distances. This provides a simple yet powerful method to dissect problems involving distances, such as the one we explored with the two cars becoming 5 miles apart.