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Burning candles

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1. Maximum burning time of a candle ●○○○○ 10 ptsA candle with an original length of $12\ \text{cm}$ burns so that its remaining length depends on the burning time as shown in the table below: | Burning time (minutes) | $10$ | $20$ | $30$ | $40$ | $50$ | $\ldots$ | |---|---|---|---|---|---|---| | Remaining length (cm) | $11.2$ | $10.4$ | $9.6$ | $8.8$ | $8$ | $\ldots$ | Assuming the remaining length decreases linearly with burning time, find the maximum number of minutes the candle can burn.
2. Time with exactly two candles burning ●●○○○ 20 ptsThree candles can burn for $30$, $40$, and $50$ minutes respectively, but they are not lit at the same time. Each candle burns until it is used up. The three candles are burning simultaneously for a total of $10$ minutes, and exactly one candle is burning for a total of $20$ minutes. Find the total time, in minutes, during which exactly two candles are burning simultaneously.
3. Candle height at half burn time ●●○○○ 20 ptsA large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. (The candle burns down each individual centimeter at a fixed rate.) Suppose it takes $T$ seconds for the candle to burn down completely. Compute the candle's height in centimeters $\tfrac{T}{2}$ seconds after it is lit.
4. Sum of initial candle lengths ●●○○○ 20 ptsTwo candles have lengths that differ by 32 cm. The longer one is lit at 15:00 and the shorter one at 19:00. At 21:00 they have the same length. The longer one burns out completely at 22:00, and the shorter one at midnight. Each candle burns at a constant rate. What is the sum of the initial lengths of the candles, in centimetres?
5. Dragon burning a village ●●●○○ 50 ptsTrodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many seconds does it take Trodgor to burn down the entire village?
6. Candle burn-out time from catch-up times ●●●○○ 50 ptsThree candles have the same length but different thicknesses, and each burns at a constant rate until it burns out completely. The first candle was lit 1 hour earlier than the other two, which were lit at the same time. At some moment the first and third candles had the same length, and 2 hours after that moment the first and second candles had the same length. The second candle burns out in 12 hours and the third in 8 hours. How many hours does it take for the first candle to burn out?

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