Problem 25 - Algebra 1: Concepts and Skills

Question

estimate that you will travel on a highway for 205 miles at an average speed of 55 miles per hour. How much time will you need for this part of the trip? Round your answer to the nearest whole hour. (A) 1 hour (B) 2 hours (C) 3 hours (D) 4 hours

Step-by-Step Solution

Answer

The time needed for this part of the trip is about 4 hours. Therefore, (D) 4 hours is the most accurate choice.

Step 1: Write down the given parameters

The distance to be covered is given as 205 miles and the average speed is provided as 55 miles per hour.

Step 2: Apply the time formula

Time is calculated by dividing the distance by the speed. Plug the given distance (205 miles) and speed (55 miles per hour) into the formula to find the time. So, \( \text{Time} = \frac{205 \text{ miles}}{55 \text{ mph}}\).

Step 3: Calculate and round the result

Divide 205 by 55 to get the time in hours. Then, round this result to the nearest whole hour. This is about 3.73 hours. Rounding to the nearest hour would give 4 hours.

Key Concepts

Distance Speed Time Formula

When it's time to solve problems involving motion, the distance speed time formula becomes your best friend. Essentially, this formula connects three critical aspects of motion – how far, how fast, and how long.

For any journey or motion, when you know two of these variables, you can find out the third. The formula is typically stated as the equation:
\[\begin{equation}\text{Distance} = \text{Speed} \times \text{Time}\end{equation}\]
That said, if the problem provides you the distance and the speed, such as in our exercise where we have a distance of 205 miles and a speed of 55 miles per hour, you can rearrange the formula to solve for time:
\[\begin{equation}\text{Time} = \frac{\text{Distance}}{\text{Speed}}\end{equation}\]

This simple reorganization allows us to plug in our known values to estimate how long a trip will take. But remember, the formula assumes that the speed is constant. If your speed fluctuates throughout your journey, calculating time might involve some additional steps to find an average speed first. Which brings us to the next important concept, average speed calculation.

Average Speed Calculation

The concept of average speed is crucial, especially when travel doesn't involve a constant velocity. Average speed isn't simply adding up two speeds and dividing by two; it's the total distance traveled divided by the total time taken.

To find it, you use the formula:
\[\begin{equation}\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\end{equation}\]
Imagine you go 30 miles in 1 hour and then another 45 miles in 1 hour. Your average speed isn't the midpoint between 30 mph and 45 mph. Instead, you add up the total distance, which is 75 miles, and divide by the total time, which is 2 hours, resulting in an average speed of 37.5 mph.

In cases where speeds are given for different intervals of the journey, you calculate the distance for each interval based on the speed and then find the total time accordingly. Sometimes students forget to consider the time factor when averaging speeds, which can lead to mistakes. Always consider the time spent traveling at each speed to get the correct average.

Math Word Problems

Math word problems often seem intimidating, but they are just real-world scenarios phrased in a narrative. Taking a strategic approach can demystify them:

• Understand the problem: Read it carefully, maybe more than once.
• Identify what's given: Look for numbers, units, and keywords that imply mathematical operations.
• Visualize the problem: If it helps, draw a diagram or a chart.
• Formulate an equation: Based on what you've identified, put together an equation or set of equations.
• Solve step-by-step: Be methodical, and don't rush. Check your math at each step.
• Verify your answer: Does it make sense in the context of the question?

The exercise presented here asked us to use information within a word problem to find a solution with a step-by-step method, which is common in math exercises. We started by identifying our 'givens' – distance and average speed – then applied the appropriate formula. Finally, we arrived at an answer after doing the calculations. Remember, practice and familiarity with common patterns in word problems can greatly boost your confidence and ability to solve them efficiently.