Due to construction, the speed limit along an 8text{-mile} section of highway is reduced from 55text{ miles per hour} to...

1. Translate the problem requirements: We need to find the difference in travel time for an 8-mile section when speed changes from 55 mph to 35 mph, with the answer in minutes.
2. Calculate travel time at original speed: Use \(\mathrm{time = distance ÷ speed}\) to find how long it takes at 55 mph.
3. Calculate travel time at reduced speed: Use the same formula to find how long it takes at 35 mph.
4. Find the time difference and convert to minutes: Subtract the two times and convert from hours to minutes to match the answer format.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're being asked to find. We have an 8-mile stretch of highway where the speed limit changes from 55 miles per hour down to 35 miles per hour due to construction. We need to figure out how much extra time this slower speed will add to our journey, and we need to give our answer in minutes.

Think of it this way: if you're driving slower, it takes longer to cover the same distance. We want to know exactly how much longer.

Process Skill: TRANSLATE - Converting the problem scenario into clear mathematical requirements

2. Calculate travel time at original speed

At the original speed of 55 mph, how long does it take to travel 8 miles?

Let's think about this in everyday terms: if you're going 55 miles in one hour, how long does it take to go just 8 miles? It would be a fraction of an hour.

We use the relationship: \(\mathrm{time = distance ÷ speed}\)

Time at 55 mph = \(\mathrm{8 \, miles ÷ 55 \, miles \, per \, hour = \frac{8}{55} \, hours}\)

Let's convert this to a decimal to make it easier to work with:
\(\mathrm{8 ÷ 55 = 0.145 \, hours}\) (approximately)

3. Calculate travel time at reduced speed

Now let's do the same calculation for the reduced speed of 35 mph.

Time at 35 mph = \(\mathrm{8 \, miles ÷ 35 \, miles \, per \, hour = \frac{8}{35} \, hours}\)

Let's simplify this fraction first: \(\mathrm{\frac{8}{35}}\) can be written as \(\mathrm{\frac{8}{35}}\)
Converting to decimal: \(\mathrm{8 ÷ 35 = 0.229 \, hours}\) (approximately)

4. Find the time difference and convert to minutes

Now we find how much extra time the slower speed adds:

Extra time = Time at 35 mph - Time at 55 mph
Extra time = \(\mathrm{\frac{8}{35} - \frac{8}{55} \, hours}\)

To subtract these fractions, let's factor out the 8:
Extra time = \(\mathrm{8 × (\frac{1}{35} - \frac{1}{55}) \, hours}\)

To subtract \(\mathrm{\frac{1}{35} - \frac{1}{55}}\), we need a common denominator. The easiest approach is to use \(\mathrm{35 × 55 = 1925}\), but let's be smarter about this.

\(\mathrm{\frac{1}{35} - \frac{1}{55} = \frac{(55 - 35)}{(35 × 55)} = \frac{20}{(35 × 55)} = \frac{20}{1925}}\)

So: Extra time = \(\mathrm{8 × (\frac{20}{1925}) = \frac{160}{1925} \, hours}\)

Let's simplify: \(\mathrm{\frac{160}{1925} = \frac{32}{385} \, hours}\)

Now convert to minutes by multiplying by 60:
Extra time in minutes = \(\mathrm{(\frac{32}{385}) × 60 = \frac{1920}{385} ≈ 4.99 \, minutes}\)

This rounds to approximately 5 minutes.

Final Answer

The extra time needed to travel the 8-mile section at the reduced speed limit is approximately 5 minutes.

Looking at our answer choices:

1. 5
2. 8
3. 10
4. 15
5. 24

Our calculated answer of approximately 5 minutes matches choice (A).

The answer is (A) 5.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what the question is asking for: Students might think they need to find the total time for each speed rather than the difference in time. This leads them to calculate times separately but forget to subtract them.

2. Unit confusion at the outset: Students may not notice that the question asks for the answer in minutes while speeds are given in miles per hour. They might plan to give their final answer in hours instead of converting to minutes.

3. Setting up the wrong relationship: Some students confuse the time-distance-speed relationship and might try to calculate \(\mathrm{speed = time × distance}\) instead of \(\mathrm{time = distance ÷ speed}\).

Errors while executing the approach

1. Fraction arithmetic errors: When subtracting \(\mathrm{\frac{8}{55} - \frac{8}{35}}\) (or \(\mathrm{\frac{1}{55} - \frac{1}{35}}\)), students often struggle with finding common denominators or make arithmetic mistakes. They might incorrectly calculate the common denominator or make errors when subtracting the numerators.

2. Decimal conversion mistakes: Students may make rounding errors when converting fractions to decimals (like \(\mathrm{\frac{8}{55}}\) and \(\mathrm{\frac{8}{35}}\)) or lose precision too early in their calculations, leading to significantly wrong final answers.

3. Unit conversion errors: Even when students remember they need minutes, they might multiply by the wrong conversion factor or forget to convert altogether, leaving their answer in hours (which would be around 0.08 hours instead of 5 minutes).

Errors while selecting the answer

1. Forgetting to round appropriately: Students might calculate 4.99 minutes but select a choice like 8 or 10 instead of recognizing that 4.99 rounds to 5. They may not realize that 'approximately' in the question means they should round to the nearest whole number.

2. Choosing based on gut feeling rather than calculation: Some students might think that reducing speed from 55 to 35 mph (a reduction of 20 mph) should result in a much larger time difference and incorrectly select choices like 15 or 24 minutes without verifying their calculation.