Let's define the problem mathematically by representing the height of each candle as a function of time.

Definitions:

• The initial height of the 12-inch candle is 12 inches, and it burns at a rate of 0.5 inches per hour.

  • After $h$ hours, the height of the 12-inch candle will be: $\text{Height of 12-inch candle} = 12 - 0.5h$

• The initial height of the 18-inch candle is 18 inches, and it burns at a faster rate of 2 inches per hour.

  • After $h$ hours, the height of the 18-inch candle will be: $\text{Height of 18-inch candle} = 18 - 2h$

Setting up the equation:

To find when the candles will be the same height, we set these two expressions equal to each other: $12 - 0.5h = 18 - 2h$

Solving the equation:

1. First, subtract $12$ from both sides: $-0.5h = 6 - 2h$

2. Add $2h$ to both sides to get all the $h$ terms on one side: $1.5h = 6$

3. Divide both sides by 1.5 to isolate $h$: $h = \frac{6}{1.5} = 4$

Answer:

The candles will be the same height after 4 hours.

Would you like more details, or do you have any questions?

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Here are 5 related questions for practice:

1. After how many hours will the 18-inch candle burn out completely?
2. What will the height of each candle be after 3 hours?
3. How much taller is the 18-inch candle than the 12-inch candle after 2 hours?
4. How many hours will it take for the 12-inch candle to burn out?
5. What if the burning rate of the 12-inch candle was increased to 1 inch per hour—when would they be the same height?

Tip: Always check if both sides of the equation balance after solving to ensure correctness.