a surveyor is 3 miles from a mountain. The angle of elevation from the ground to the top of the mountain is 15 degrees. What is the height of the mountain?
0.8
Why did the surveyor bring a ladder to measure the mountain? Because they wanted to climb to new heights in their career!
Now, let's use a little math to solve this. We have a right triangle formed by the surveyor, the top of the mountain, and a point on the mountain directly below the top. The distance from the surveyor to the mountain is the base of the triangle, and the height of the mountain is the side opposite to the 15-degree angle.
Using trigonometry, we can determine the height h by using the tangent function:
tan(15°) = h/3
By rearranging the equation, we find:
h = 3 * tan(15°)
Calculating that gives us an approximate height of:
h ≈ 0.77 miles
So, the height of the mountain is approximately 0.77 miles. Keep in mind that this is not accounting for any other factors such as dips or slopes on the mountain—just a basic calculation based on the given information!
To find the height of the mountain, we can use trigonometry. The angle of elevation forms a right triangle, with the base being the distance from the surveyor to the mountain (3 miles) and the height being the height of the mountain, which we want to find.
We can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle. In this case, the height of the mountain is the opposite side and the distance from the surveyor is the adjacent side.
Using the formula: tangent(angle) = opposite/adjacent, we can rearrange the formula to solve for the height.
tangent(15 degrees) = height/3 miles
Now let's calculate the height:
Taking the tangent of 15 degrees, we have:
tan(15 degrees) ≈ 0.26795
So the equation becomes:
0.26795 = height/3 miles
To solve for the height, we can cross-multiply:
height = 0.26795 * 3 miles
height ≈ 0.80385 miles
Therefore, the height of the mountain is approximately 0.80385 miles.
Assuming the 3 miles is the horizontal distance to the "inside" of the mountain, this is a basic tangent ratio
tan15 = height/3
height = 3tan15° = ... miles