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## Homework Statement

In the problem, we are to consider two candles, call them C1 and C2, with different heights and different thicknesses. Call the height of C1 H1, and for C2, call it H2. The taller candle burns can burn for 7/2 hours, and the short one, 5 hours. After two hours lapses, the candles have equal heights. Two hours ago, what fraction of the candle's height gave the short candle's height?

## Homework Equations

## The Attempt at a Solution

Here some observations, of which I am not certain are true, that I made. Assuming the candles burn at a constant rate, the rate of C1 is H1/3.5, and the rate of C2 is H2/5.

After two hours of burning has lapsed, h1/h2 = 1.

Now, I am going to assume the candles are cylindrical.

[itex]V_1 = \pi r^2_1H_1[/itex] I am also going to assume that the burning process is such that the radius is not altered.

[itex]\frac{dV_1}{dt} = \pi r^2_1 \frac{dH_1}{dt}[/itex]. I know how the height changes with time--it's constant. [itex]\frac{dH_1}{dt} = \frac{H_1}{3.5}[/itex] If I write it in the form [itex]\frac{dH_1}{dt} = (H_1)(3.5)^{-1}[/itex], this is a differential equation in which I can use the separation of variables method.

[itex]\int H_1 dH_1 = (3.5)^{-1} \int dt[/itex]

which comes out to be [itex]H_1 = \sqrt{\frac{t}{3.5} + C}[/itex]

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There are a few details I am confused about. Did I properly apply the separation of variables method? If so, what does the constant C represent, and can it be set to zero? Will this solution lead anywhere?

Thank you, and please don't give every detail of the problem away.

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