How long does it take a 19 kW steam engine to do \( 6.8 \times 10^{7} \mathrm{~J} \) of work? \( (\mathrm{k}=\mathrm{kilo}=1000) \)

To find out how long it takes for a 19 kW steam engine to do \( 6.8 \times 10^{7} \) J of work, we can use the formula: \[ \text{Power (P)} = \frac{\text{Work (W)}}{\text{Time (t)}} \] Rearranging gives us: \[ t = \frac{W}{P} \] Substituting the values: \[ t = \frac{6.8 \times 10^{7} \text{ J}}{19 \times 10^{3} \text{ W}} \\ t \approx \frac{6.8 \times 10^{7}}{19000} \\ t \approx 3587.37 \text{ seconds} \] So, it would take approximately 3587 seconds, or about 59.8 minutes, for the engine to perform that amount of work. For historical enthusiasts, the steam engine revolutionized industries in the 18th and 19th centuries, fueling not just factories but also railways and ships, ushering in the Industrial Revolution—an era that reshaped economies and social structures globally! If you're ever in a jam while tackling such problems, remember to double-check your unit conversions, as a small mistake can lead to big discrepancies in your answer. Always ensure you're working with consistent units—like Watts with Joules and seconds—to keep your calculations accurate and flowing smoothly!