41. A piece of wood burns completely in $1 \mathrm{~s}$ at $600^{\circ} \mathrm{C}$. The time it takes for the wood to burn is doubled for every $10^{\circ} \mathrm{C}$ drop in temperature and halved for every $10^{\circ} \mathrm{C}$ increase in temperature. Determine how long the piece of wood would take to burn at each temperature. a) $500^{\circ} \mathrm{C}$ b) $650^{\circ} \mathrm{C}$

Step 1 ：Given that a piece of wood burns completely in 1 second at \(600^\circ\mathrm{C}\), and the time it takes for the wood to burn is doubled for every \(10^\circ\mathrm{C}\) drop in temperature and halved for every \(10^\circ\mathrm{C}\) increase in temperature.

Step 2 ：Let's find the time it takes for the wood to burn at \(500^\circ\mathrm{C}\) and \(650^\circ\mathrm{C}\).

Step 3 ：For \(500^\circ\mathrm{C}\), the temperature dropped by \(100^\circ\mathrm{C}\) from \(600^\circ\mathrm{C}\). Since the time doubles for every \(10^\circ\mathrm{C}\) drop, the time will double 10 times: \(1\times 2^{10} = 1024\) seconds.

Step 4 ：For \(650^\circ\mathrm{C}\), the temperature increased by \(50^\circ\mathrm{C}\) from \(600^\circ\mathrm{C}\). Since the time halves for every \(10^\circ\mathrm{C}\) increase, the time will halve 5 times: \(1\times \frac{1}{2^5} = 0.03125\) seconds.

Step 5 ：\(\boxed{\text{a) At } 500^\circ\mathrm{C}, \text{ the piece of wood would take } 1024 \text{ seconds to burn completely.}}\)

Step 6 ：\(\boxed{\text{b) At } 650^\circ\mathrm{C}, \text{ the piece of wood would take } 0.03125 \text{ seconds to burn completely.}}\)