To determine how long it takes for a steam engine with a power of [tex]\(1.51 \times 10^4 \, \text{W}\)[/tex] to perform [tex]\(8.72 \times 10^6 \, \text{J}\)[/tex] of work, we need to use the formula that relates power, work, and time. The relationship is given by:
[tex]\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \][/tex]
Rearranging this equation to solve for time, we get:
[tex]\[ \text{Time} = \frac{\text{Work}}{\text{Power}} \][/tex]
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]
By performing the division:
[tex]\[ \text{Time} = \frac{8.72 \times 10^6}{1.51 \times 10^4} \][/tex]
Carrying out the division:
[tex]\[ \text{Time} = 577.4834437086092 \, \text{s} \][/tex]
Rounding to three significant figures, we have:
[tex]\[ \text{Time} \approx 577.483 \, \text{s} \][/tex]
Thus, among the given options, the correct answer is:
[tex]\[ 5.77 \times 10^2 \, \text{s} \][/tex]
