Question

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

Short 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-by-step solution

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.

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: \[344 * \frac{1 mile}{1609.34}\ \text{in}\ \frac{miles}{s}\]

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: \[\left (344 * \frac{1 mile}{1609.34} \right ) * \frac{3600}{1}\ \text{in}\ \frac{miles}{hour}\]

Calculate

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

Round the result

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