How Long Does It Take A $1.51 \times 10^4 , \text{W}$ Steam Engine To Do $8.72 \times 10^6 , \text{J}$ Of Work? Round Your Answer To Three Significant Figures.A. 1.02 × 10 1 S 1.02 \times 10^1 \, \text{s} 1.02 × 1 0 1 S B. $5.77 \times 10^2

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Understanding the Problem

In this problem, we are given a steam engine with a power output of $1.51 \times 10^4 , \text{W}$ and a work requirement of $8.72 \times 10^6 , \text{J}$. We need to determine the time it takes for the steam engine to complete this work. To solve this problem, we will use the formula for power, which is the rate at which work is done.

Power and Work

Power is defined as the rate at which work is done, and it is measured in watts (W). The formula for power is:

P=WtP = \frac{W}{t}

where $P$ is the power, $W$ is the work, and $t$ is the time.

Rearranging the Formula

To solve for time, we need to rearrange the formula to isolate $t$. We can do this by multiplying both sides of the equation by $t$, which gives us:

Pt=WPt = W

Solving for Time

Now that we have the formula rearranged, we can plug in the values given in the problem. We know that the power is $1.51 \times 10^4 , \text{W}$ and the work is $8.72 \times 10^6 , \text{J}$. Plugging these values into the formula, we get:

1.51×104 W×t=8.72×106 J1.51 \times 10^4 \, \text{W} \times t = 8.72 \times 10^6 \, \text{J}

Simplifying the Equation

To simplify the equation, we can divide both sides by $1.51 \times 10^4 , \text{W}$. This gives us:

t=8.72×106 J1.51×104 Wt = \frac{8.72 \times 10^6 \, \text{J}}{1.51 \times 10^4 \, \text{W}}

Evaluating the Expression

Now that we have the equation simplified, we can evaluate the expression to find the value of $t$. We can do this by dividing the numerator by the denominator:

t=8.72×106 J1.51×104 W=5.77×102 st = \frac{8.72 \times 10^6 \, \text{J}}{1.51 \times 10^4 \, \text{W}} = 5.77 \times 10^2 \, \text{s}

Rounding the Answer

Finally, we need to round our answer to three significant figures. The value of $t$ is $5.77 \times 10^2 , \text{s}$, which is already rounded to three significant figures.

Conclusion

In this problem, we used the formula for power to determine the time it takes for a steam engine to complete a certain amount of work. We rearranged the formula to isolate $t$, plugged in the values given in the problem, and evaluated the expression to find the value of $t$. Our final answer is $5.77 \times 10^2 , \text{s}$.

Answer

Q: What is the relationship between power and work?

A: The relationship between power and work is that power is the rate at which work is done. The formula for power is:

P=WtP = \frac{W}{t}

where $P$ is the power, $W$ is the work, and $t$ is the time.

Q: How do I calculate the time it takes for a steam engine to complete a certain amount of work?

A: To calculate the time it takes for a steam engine to complete a certain amount of work, you can use the formula:

t=WPt = \frac{W}{P}

where $t$ is the time, $W$ is the work, and $P$ is the power.

Q: What is the unit of power?

A: The unit of power is the watt (W).

Q: What is the unit of work?

A: The unit of work is the joule (J).

Q: How do I convert watts to joules per second?

A: To convert watts to joules per second, you can use the following conversion factor:

1 W=1 J/s1 \, \text{W} = 1 \, \text{J/s}

Q: What is the significance of significant figures in scientific calculations?

A: Significant figures are a way to express the precision of a measurement or calculation. In scientific calculations, it is essential to round answers to the correct number of significant figures to avoid errors.

Q: How do I round a number to three significant figures?

A: To round a number to three significant figures, you need to look at the fourth significant figure. If it is less than 5, you round down. If it is 5 or greater, you round up.

Q: What is the final answer to the problem of how long it takes a steam engine to do work?

A: The final answer to the problem of how long it takes a steam engine to do work is $5.77 \times 10^2 , \text{s}$.

Q: What is the relationship between the power of a steam engine and the time it takes to complete a certain amount of work?

A: The relationship between the power of a steam engine and the time it takes to complete a certain amount of work is that the power is inversely proportional to the time. This means that if the power increases, the time it takes to complete the work decreases.

Q: How do I use the formula for power to solve problems involving steam engines?

A: To use the formula for power to solve problems involving steam engines, you need to plug in the values for power, work, and time into the formula:

P=WtP = \frac{W}{t}

and solve for the unknown variable.

Q: What are some real-world applications of steam engines?

A: Some real-world applications of steam engines include:

  • Power generation
  • Transportation (e.g., steam locomotives)
  • Industrial processes (e.g., textile manufacturing)
  • Heating and cooling systems

Q: What are some limitations of steam engines?

A: Some limitations of steam engines include:

  • Low efficiency
  • High maintenance costs
  • Limited power output
  • Environmental concerns (e.g., greenhouse gas emissions)

Q: How do I choose the right steam engine for a particular application?

A: To choose the right steam engine for a particular application, you need to consider factors such as:

  • Power requirements
  • Efficiency
  • Maintenance costs
  • Environmental concerns
  • Space constraints