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question:How does the fact that light from lightning reaches you before its sound prove the difference in speed between light and sound, and what method can be used to estimate the speed of light using this observation?
answer:The phenomenon of seeing the lightning flash before hearing the thunder demonstrates that light travels significantly faster than sound in a medium. The speed of light is vastly greater than the speed of sound, which in air is approximately 343 meters per second. To estimate the speed of light using this effect, you can follow these steps: 1. Observe a lightning flash and start a stopwatch immediately. 2. Count the time (in seconds) until you hear the thunder. This time interval represents the difference in travel time between light and sound. 3. Since sound travels at a known speed of 343 m/s, you can calculate the distance traveled by sound during that time interval using the formula: {eq}{rm{Distance}};{rm{(m)}} = {rm{Speed}};{rm{(m/s)}} times {rm{Time}};{rm{(sec)}}{/eq} 4. The distance traveled by sound is also the approximate distance to the lightning strike. 5. Now, divide the distance by the time it took for the light to reach you to estimate the speed of light: {eq}{rm{Speed}};{rm{(m/s)}} = dfrac{{{rm{Distance}};{rm{(m)}}}}{{{rm{Time}};{rm{(sec)}}}}{/eq} Keep in mind that this method provides a rough estimate, as it doesn't account for the curvature of the Earth or atmospheric conditions affecting the speed of light and sound. The actual speed of light in a vacuum is approximately 299,792,458 meters per second.
question:Prove that every integer greater than 1 can be written as the product of an odd natural number and a power of 2.
answer:Consider the largest power of 2 that divides the integer. If this is 2^k, then the integer can be written as 2^k cdot n, where n is the integer divided by 2^k. Since 2^k is even, n must be odd. Therefore, the integer can be written as the product of an odd natural number and a power of 2.
question:Who are credited with creating the first gasoline-powered automobile in the United States, Charles Edgar Duryea and his brother or someone else?
answer:Charles Edgar Duryea and his brother are credited with creating the first gasoline-powered automobile in the United States. Their invention marked a significant milestone in the history of automotive engineering.
question:Simplify the expression left(frac{5 left(-cos left(frac{7 pi }{90}right)+i sin left(frac{7 pi }{90}right)right)}{sqrt{2}}right)^5.
answer:The expression simplifies to frac{3125 left(-sin left(frac{7 pi }{18}right)+i cos left(frac{7 pi }{18}right)right)}{4 sqrt{2}} after using De Moivre's Theorem and simplifying the trigonometric functions.