Step 1: Understanding the Concept:
Efficiency of a machine (like an engine) is the ratio of the useful output power to the total input power. It is usually expressed as a percentage.
Step 2: Key Formula or Approach:
The formula for efficiency ($\eta$) is:
\[ \eta = \frac{\text{Output Power}}{\text{Input Power}} \]
We are given the output power and the efficiency, and we need to find the input power. We can rearrange the formula:
\[ \text{Input Power} = \frac{\text{Output Power}}{\eta} \]
Step 3: Detailed Explanation:
Given:
- Output Power = 1000 W.
- Efficiency, $\eta = 80% = \frac{80}{100} = 0.8$.
Using the rearranged formula:
\[ \text{Input Power} = \frac{1000}{0.8} \]
\[ \text{Input Power} = \frac{1000}{8/10} = \frac{1000 \times 10}{8} = \frac{10000}{8} \]
\[ \text{Input Power} = 1250 \text{ W} \]
Step 4: Final Answer:
The input power required by the engine is 1250 W. Therefore, option (C) is correct.