A mountaintop is a height y above the level ground. A woman measures the angle of elevation of the mountaintop to be θ when she is a horizontal distance x from the mountaintop. After walking a distance d closer to the mountain, she measures the angle of elevation of the mountaintop to be φ. Neglecting the height of the woman’s eyes above the ground, find an expression for the height of the mountain, y, in terms of d, φ, and θ. Note that your expression cannot contain x.
y = (180-(180-θ-φ)(φ)(θ))sinθ
y=-d/[1-(tanφ/tanθ)]
i forgot something in the other answer oops.
y=-d*tanφ/[1-(tanφ/tanθ)]
this answer works
To find an expression for the height of the mountain, y, in terms of d, φ, and θ, we can use trigonometry. Let's use the right triangle created by the woman, the mountaintop, and the point on the ground directly below the mountaintop.
From the given information, when the woman is at a horizontal distance x from the mountaintop, the angle of elevation is θ. This means that the opposite side of the triangle is y, and the adjacent side is x.
Using trigonometry, we can express the tangent of the angle θ as the ratio of the opposite side to the adjacent side:
tan(θ) = y/x
Simplifying this equation, we get:
y = x * tan(θ)
Now, let's consider the situation after the woman walks a distance d closer to the mountain. The new angle of elevation is φ.
In the new right triangle, the adjacent side is (x - d) since the woman has walked a distance d closer to the mountain. The opposite side is still y, as the height of the mountain remains the same.
Using trigonometry again, we can express the tangent of the angle φ as the ratio of the opposite side to the adjacent side:
tan(φ) = y/(x - d)
Rearranging this equation, we can solve for y:
y = (x - d) * tan(φ)
Therefore, the expression for the height of the mountain, y, in terms of d, φ, and θ is:
y = (x - d) * tan(φ)