To determine the time it takes for a steam engine with a power output of [tex]\(1.51 \times 10^4\)[/tex] watts to do [tex]\(8.72 \times 10^6\)[/tex] joules of work, we need to use the relationship between work, power, and time. The relevant formula to use is:
[tex]\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \][/tex]
Solving for time:
[tex]\[ \text{Time} = \frac{\text{Work}}{\text{Power}} \][/tex]
1. Identify the given values:
- Power ([tex]\(P\)[/tex]): [tex]\(1.51 \times 10^4\)[/tex] W
- Work ([tex]\(W\)[/tex]): [tex]\(8.72 \times 10^6\)[/tex] J
2. Substitute the given values into the equation:
[tex]\[ \text{Time} = \frac{8.72 \times 10^6 \, \text{J}}{1.51 \times 10^4 \, \text{W}} \][/tex]
3. Compute the division:
[tex]\[ \text{Time} = \frac{8.72 \times 10^6}{1.51 \times 10^4} \][/tex]
[tex]\[ \text{Time} \approx 577.483 \, \text{s} \][/tex]
4. Round the answer to three significant figures:
[tex]\[ \text{Time} \approx 577 \, \text{s} \][/tex]
Therefore, the time it takes for the steam engine to do the work is approximately [tex]\(5.77 \times 10^2\)[/tex] seconds.
So the correct answer is:
[tex]\[ \boxed{5.77 \times 10^2 \, \text{s}} \][/tex]
