Problem 10 - Chemistry: The Central Science

Question

Show the steps to convert the speed of sound, 344 meters per second, into miles per hour. [Section 1.6]

Step-by-Step Solution

Answer

To convert the speed of sound (344 m/s) into miles per hour, first convert meters to miles by multiplying with the conversion factor \(\frac{1 mile}{1609.34 meters}\). Then, convert seconds to hours by multiplying with the conversion factor \(\frac{3600 seconds}{1 hour}\). Finally, calculate and round the result: \[\left (344 * \frac{1 mile}{1609.34} \right ) * \frac{3600}{1} \approx 767.27\ \text{miles per hour}\].

Step 1: Understand the problem and units conversion

We are given the speed of sound in meters per second (m/s) and asked to convert it to miles per hour (mph). To do this, we need to use conversion factors to convert meters to miles and seconds to hours.

Step 2: Convert meters to miles

There are 1609.34 meters in 1 mile. To convert meters to miles, use the conversion factor: \[1 mile = 1609.34 meters\]. To perform this conversion, multiply the speed in meters per second by the conversion factor: \[344 \frac{m}{s} * \frac{1 mile}{1609.34 meters}\] The "meters" unit in the numerator and denominator cancels out, leaving us with: \[344 * \frac{1 mile}{1609.34}\ \text{in}\ \frac{miles}{s}\]

Step 3: Convert seconds to hours

There are 3600 seconds in 1 hour. To convert seconds to hours, use the conversion factor: \[1 hour = 3600 seconds\]. To perform this conversion, multiply the speed in miles per second by the conversion factor: \[\left (344 * \frac{1 mile}{1609.34} \right )\frac{1}{s} * \frac{3600 s}{1 hour}\] The "seconds" unit in the numerator and denominator cancels out, now we have remaining units as miles per hour: \[\left (344 * \frac{1 mile}{1609.34} \right ) * \frac{3600}{1}\ \text{in}\ \frac{miles}{hour}\]

Step 4: Calculate

Multiply the values together to find the speed in miles per hour: \[(344 * \frac{1}{1609.34}) * 3600\] \[=\ 767.269\ \text{in}\ \frac{miles}{hour}\]

Step 5: Round the result

To obtain a reasonable level of precision, round the result to two decimal places: \[speed\ of\ sound = 767.27\ \text{miles per hour}\]