To determine how much time is needed for a [tex]$15,000$[/tex]-watt ([tex]$W$[/tex]) engine to do [tex]$1,800,000$[/tex] joules ([tex]$J$[/tex]) of work, we need to use the formula for power:
[tex]\[ P = \frac{W}{t} \][/tex]
where [tex]\( P \)[/tex] is the power in watts, [tex]\( W \)[/tex] is the work in joules, and [tex]\( t \)[/tex] is the time in seconds.
Here we are given:
- Power [tex]\( P \)[/tex] is [tex]$15,000$[/tex] watts
- Work [tex]\( W \)[/tex] is [tex]$1,800,000$[/tex] joules
We need to find the time [tex]\( t \)[/tex]. We can rearrange the power formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{W}{P} \][/tex]
Substitute the given values into the formula:
[tex]\[ t = \frac{1,800,000 \, J}{15,000 \, W} \][/tex]
Perform the division:
[tex]\[ t = 120 \, s \][/tex]
Therefore, the time needed for a [tex]$15,000$[/tex]-watt engine to do [tex]$1,800,000$[/tex] joules of work is [tex]\( \boxed{120 \, s} \)[/tex].
