Final answer:
It takes a 19 kW steam engine approximately 47 minutes to do 5.4 × 10⁷ J of work, using the formula t = W/P and converting kilowatts to joules per second.
Step-by-step explanation:
To determine how long it takes a 19 kW (kilowatt) steam engine to do 5.4 × 10⁷ J (joules) of work, we must apply the relationship between power, work, and time. Power is defined as the rate of doing work or energy transfer per unit time. The formula that relates power (P) with work (W) and time (t) is P = W/t, which can be rearranged to solve for time: t = W/P.
Plugging in the given values, we have t = (5.4 × 10⁷ J) / (19 kW), and since 1 kW = 1,000 J/s, we convert 19 kW to joules per second (19,000 J/s). Therefore, t = (5.4 × 10⁷ J) / (19,000 J/s), which yields t = 2,842.105 s or approximately 47.368 minutes. So, it takes roughly 47 minutes for the steam engine to complete the work.