From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the hill is 100 meters high, how far is the point from the base of the hill?
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From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the hill is 100 meters high, how far is the point from the base of the hill?
Q: From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the hill is 100 meters high, how far is the point from the base of the hill?
Step 1: Understand the problem. We have a hill that is 100 meters high and we want to find out how far away we are from the base of the hill.
Step 2: Identify the angle of elevation. The angle of elevation to the top of the hill is 30 degrees.
Step 3: Recall the relationship between height, distance, and angle in a right triangle. We can use the tangent function, which is defined as the opposite side (height of the hill) over the adjacent side (distance from the base).
Step 4: Write the formula for tangent: tan(angle) = opposite / adjacent. In our case, tan(30 degrees) = height / distance.
Step 5: Substitute the known values into the formula. We know the height is 100 meters and the angle is 30 degrees, so we have: tan(30 degrees) = 100 / distance.
Step 6: Find the value of tan(30 degrees). It is equal to 1/√3.
Step 7: Set up the equation: 1/√3 = 100 / distance.
Step 8: Rearrange the equation to solve for distance: distance = 100 / (1/√3).
Step 9: Simplify the equation: distance = 100 * √3.
Step 10: Calculate the final distance: distance = 100√3 meters.
