The earth travels around the sun at a speed of approximately 18.5 text{ miles per second}. The approximate speed is...

1. Translate the problem requirements: We need to convert Earth's speed from \(\mathrm{18.5\ miles\ per\ second}\) to miles per hour. This requires understanding that we need to find how many miles Earth travels in one hour given its per-second speed.
2. Identify the conversion relationship: Determine how many seconds are in one hour to establish the multiplication factor needed for conversion.
3. Apply unit conversion logic: Multiply the per-second speed by the number of seconds in an hour to get the per-hour speed.
4. Evaluate against answer choices: Compare our calculated result with the given options, using the magnitude of answer choices to guide appropriate precision in calculations.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're being asked to do in everyday terms. We know that Earth travels at \(\mathrm{18.5\ miles}\) every single second. Think about how fast that is - in just one second, Earth covers \(\mathrm{18.5\ miles}\)! Now we want to figure out: if Earth keeps going at this same speed for an entire hour, how many miles would it cover in total?

This is like asking: if a car travels \(\mathrm{30\ miles}\) every hour, how far does it go in \(\mathrm{3\ hours}\)? We'd multiply \(\mathrm{30 \times 3 = 90\ miles}\). Here, we're doing something similar - we know the distance per second, and we want the distance per hour.

Process Skill: TRANSLATE - Converting the speed conversion question into a clear multiplication problem

2. Identify the conversion relationship

To solve this, we need to figure out how many seconds are packed into one hour. Let's think through this step by step:

• In \(\mathrm{1\ minute}\), there are \(\mathrm{60\ seconds}\)
• In \(\mathrm{1\ hour}\), there are \(\mathrm{60\ minutes}\)

So in \(\mathrm{1\ hour}\), the total number of seconds is: \(\mathrm{60\ seconds\ per\ minute \times 60\ minutes\ per\ hour = 3,600\ seconds}\)

This means that in one hour, there are exactly \(\mathrm{3,600}\) individual seconds. Since Earth travels \(\mathrm{18.5\ miles}\) in each of those seconds, we'll need to multiply \(\mathrm{18.5}\) by \(\mathrm{3,600}\) to get the total distance in one hour.

3. Apply unit conversion logic

Now we can calculate Earth's speed in miles per hour using simple multiplication:

Speed in miles per hour = \(\mathrm{18.5\ miles\ per\ second \times 3,600\ seconds\ per\ hour}\)

Let's make this multiplication easier by breaking it down:
\(\mathrm{18.5 \times 3,600}\)

I can rewrite \(\mathrm{3,600}\) as \(\mathrm{36 \times 100}\) to make the calculation simpler:
\(\mathrm{18.5 \times 36 \times 100}\)

First, let's calculate \(\mathrm{18.5 \times 36}\):
\(\mathrm{18.5 \times 36 = 18.5 \times (30 + 6)}\)
\(\mathrm{= (18.5 \times 30) + (18.5 \times 6)}\)
\(\mathrm{= 555 + 111}\)
\(\mathrm{= 666}\)

Now multiply by \(\mathrm{100}\):
\(\mathrm{666 \times 100 = 66,600}\)

Therefore, Earth travels at approximately \(\mathrm{66,600\ miles\ per\ hour}\).

4. Evaluate against answer choices

Let's check our calculated result of \(\mathrm{66,600\ miles\ per\ hour}\) against the given options:

1. \(\mathrm{1,080}\) - This is far too small
2. \(\mathrm{1,160}\) - This is also too small
3. \(\mathrm{64,800}\) - This is close but still smaller than our answer
4. \(\mathrm{66,600}\) - This matches our calculation exactly!
5. \(\mathrm{3,996,000}\) - This is way too large

Our calculation of \(\mathrm{66,600\ miles\ per\ hour}\) matches answer choice (D) perfectly.

Final Answer

The correct answer is (D) \(\mathrm{66,600\ miles\ per\ hour}\). We converted Earth's speed from \(\mathrm{18.5\ miles\ per\ second}\) to miles per hour by multiplying by the number of seconds in an hour (\(\mathrm{3,600}\)), giving us \(\mathrm{18.5 \times 3,600 = 66,600\ miles\ per\ hour}\).

Common Faltering Points

Errors while devising the approach

1. Confusing the conversion direction: Students may try to divide \(\mathrm{18.5}\) by \(\mathrm{3,600}\) instead of multiplying, thinking they need to make the number smaller when converting from a smaller unit (seconds) to a larger unit (hours). This conceptual error stems from not understanding that when you have a rate "per second" and want "per hour," you're asking how much distance accumulates over the longer time period.

2. Using incorrect time conversion factors: Students might use \(\mathrm{60}\) instead of \(\mathrm{3,600}\), forgetting that an hour contains \(\mathrm{60\ minutes}\) AND each minute contains \(\mathrm{60\ seconds}\). They may only account for one level of the conversion (either minutes to seconds OR hours to minutes) rather than the complete hours to seconds conversion.

Errors while executing the approach

1. Arithmetic errors in multiplication: When calculating \(\mathrm{18.5 \times 3,600}\), students often make computational mistakes. Common errors include incorrectly calculating \(\mathrm{18.5 \times 36 = 555}\) (instead of the correct \(\mathrm{666}\)) or making errors when applying the distributive property \(\mathrm{18.5 \times (30 + 6)}\).

2. Decimal and place value errors: Students may correctly calculate \(\mathrm{18.5 \times 36 = 666}\) but then make mistakes when multiplying by \(\mathrm{100}\), either forgetting to add the zeros or incorrectly placing the decimal point, leading to answers like \(\mathrm{6,660}\) or \(\mathrm{6,66,000}\).

Errors while selecting the answer

1. Choosing a close but incorrect answer: Students who get an approximate result around \(\mathrm{64,000-67,000}\) might select choice (C) \(\mathrm{64,800}\) thinking their calculation was "close enough" rather than double-checking their work to confirm the exact answer of \(\mathrm{66,600}\) in choice (D).