Accounting for heating losses: If 2 kWh of energy is required to heat a bucket of water to a target temperature and the heat losses are 25%, what electrical energy must be supplied?

Introduction / Context:Practical heating systems are not perfectly efficient. Part of the input electrical energy is lost to the environment. This problem checks your ability to compute the necessary input energy given a required useful energy output and a specified fractional loss.

Given Data / Assumptions:

• Useful (required) thermal energy to water: 2 kWh.
• Heat losses: 25% of input energy.
• Single heating step with constant loss fraction.

Concept / Approach:

Define efficiency η as the fraction of input energy converted to useful heating. If the losses are 25%, then η = 1 − 0.25 = 0.75. The input energy E_in must satisfy E_useful = η * E_in. Rearranging gives E_in = E_useful / η.

Step-by-Step Solution:

Verification / Alternative check:

Check by forward multiplication: 2.67 kWh * 0.75 = 2.0025 kWh ≈ 2 kWh (rounding explains the small difference). This confirms consistency within rounding tolerance.

Why Other Options Are Wrong:

• 3 kWh assumes η = 2/3 (loss 33.3%), not given.
• 2.5 kWh corresponds to η = 0.8 (loss 20%).
• 3.5 kWh implies η ≈ 0.571 (loss 42.9%).
• 2 kWh ignores losses (η = 1), contradicting the problem.

Common Pitfalls:

• Mistaking loss percentage as applied to useful energy rather than input energy; correct relation uses efficiency.

Final Answer:

2.67 kWh