Cameron had two candles, one of which was 32 cm longer than the other. She lit the taller candle at 3pm and lit the shorter candle at 7pm, they were both the same height at 9pm. The taller one was completely burned out at 10pm and the shorter on was completely burned at midnight. The two candles burned at different, but constant, rates. What were the original heights of the two candles?
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Cameron has two candles with different heights. The taller candle is 32 centimeters longer than the shorter one. She lights them at different times and they burn at constant but different rates. We need to find their original heights.
Let's define our variables clearly. Let H-t be the original height of the taller candle, and H-s be the original height of the shorter candle. From the problem, we know that the taller candle is 32 centimeters longer than the shorter one, giving us our first equation: H-t equals H-s plus 32.
Now let's establish the burning rates and timeline. The taller candle burns for 7 hours total, from 3pm to 10pm, so its rate is H-t divided by 7 centimeters per hour. The shorter candle burns for 5 hours, from 7pm to midnight, so its rate is H-s divided by 5 centimeters per hour. The key moment is 9pm when both candles have the same height.
Now let's calculate the heights at 9pm. The taller candle has been burning for 6 hours, so its remaining height is H-t minus 6 times its rate, which simplifies to H-t divided by 7. The shorter candle has been burning for 2 hours, so its remaining height is H-s minus 2 times its rate, which simplifies to 3 H-s divided by 5. Since they're equal at 9pm, we get our second equation.
Now let's solve our system of equations. We have H-t equals H-s plus 32, and H-t over 7 equals 3 H-s over 5. Substituting the first equation into the second gives us H-s plus 32 over 7 equals 3 H-s over 5. Cross multiplying and simplifying, we get 160 equals 16 H-s, so H-s equals 10 centimeters. Therefore, H-t equals 42 centimeters. The original heights were 42 centimeters for the taller candle and 10 centimeters for the shorter candle.