Math Puzzles: When Will the Candles Reach the Same Height?

As a Google SEOer, I've noticed that many search queries involve interesting mathematical puzzles to test your analytical skills. One such intriguing puzzle involves two candles with different heights and burning rates. Let's dive into the puzzle and explore the mathematical concepts that can help solve it.

The Puzzle:

Two candles are present; one is 50 cm high and can burn for 3 hours, while the other is 70 cm high and can burn for 6 hours. The question is: when will these two candles reach the same height?

Breaking Down the Problem:

First, let's establish the burning rates of the two candles:
- The 50 cm candle: burns to 0 cm in 3 hours.
- The 70 cm candle: burns to 0 cm in 6 hours.

To find the heights at various times, we need to calculate the burning rates per hour:

The burning rate of the 50 cm candle is (50 - 0) cm / 3 hours 16.67 cm/hour. The burning rate of the 70 cm candle is (70 - 0) cm / 6 hours 11.67 cm/hour.

Now, let's set up the equations for the height of each candle over time:

Height of the 50 cm candle (h1) 50 - 16.67t, where t is the time in hours. Height of the 70 cm candle (h2) 70 - 11.67t.

To find when the two candles will be at the same height, we set h1 h2 and solve for t:

50 - 16.67t 70 - 11.67t

Move the terms with t to one side and the constants to the other:

50 - 70 16.67t - 11.67t

-20 5t

t 4

Thus, after 4 hours, both candles will be at the same height. We can verify this by plugging t 4 back into our equations:

Height of the 50 cm candle: 50 - 16.67*4 50 - 66.68 -16.68 (This indicates the candle is no longer burning.) Height of the 70 cm candle: 70 - 11.67*4 70 - 46.68 23.32 cm.

Alternative Interpretation:

Another way to look at the problem is to consider the time when the 50 cm candle burns out and the 70 cm candle is still burning:

The 50 cm candle burns out in 3 hours. The 70 cm candle burns out in 6 hours.

To reach the same height, the 50 cm candle will have burned for 3 hours, and the 70 cm candle will have burned for 4.5 hours (since (6 - 1.5 4.5) when the 50 cm candle is no longer burning).

Conclusion:

The key takeaway is that the two candles will be at the same height after 4 hours. This puzzle highlights the importance of understanding burning rates and how to set up and solve equations. It also emphasizes the logic behind interpreting the problem.

Remember, solving math puzzles can be fun and engaging. They help improve your analytical skills and broaden your understanding of mathematical concepts.