Galileo wanted to release a wooden ball and an iron ball from a height of 150 meters and measure the duration of their fall. He found a plane with an incline of 15 degrees that he could climb until he gets to an altitude of 150m. How far should Galileo walk up the inclined plane? Round your final answer to the nearest hundredth. .
Question
Galileo wanted to release a wooden ball and an iron ball from a height of 150 meters and measure the duration of their fall. He found a plane with an incline of 15 degrees that he could climb until he gets to an altitude of 150m. How far should Galileo walk up the inclined plane? Round your final answer to the nearest hundredth.
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Solution
To solve this problem, we need to use some trigonometry. The situation described forms a right triangle, where the altitude of 150m is the opposite side, the inclined plane is the hypotenuse, and the angle of inclination is 15 degrees.
We know that the sine of an angle in a right triangle is equal to the opposite side divided by the hypotenuse. So, we can set up the following equation:
sin(15 degrees) = 150m / hypotenuse
We can solve this equation for the hypotenuse (which represents the distance Galileo should walk up the inclined plane):
hypotenuse = 150m / sin(15 degrees)
Using a calculator, we find that sin(15 degrees) is approximately 0.2588190451. So:
hypotenuse = 150m / 0.2588190451 ≈ 579.35m
So, Galileo should walk approximately 579.35 meters up the inclined plane. Rounding to the nearest hundredth, this is 579.35 meters.
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