Question

1. -/1 points a woman stands at a horizontal distance x from a mountain and measures the angle of elevation of the mountaintop above the horizontal as θ. after walking a distance d closer to the mountain on level ground, she finds the angle to be φ. find a general equation for the height y of the mountain in terms of d, φ, and θ, neglecting the height of her eyes above the ground.

Explanation:

Step1: Use tangent function for first - position

Let the initial horizontal distance be $x$ and the height of the mountain be $y$. We know that $\tan\theta=\frac{y}{x}$, so $y = x\tan\theta$.

Step2: Use tangent function for second - position

After walking a distance $d$ closer to the mountain, the new horizontal distance is $x - d$ and the angle of elevation is $\varphi$. So $\tan\varphi=\frac{y}{x - d}$, which implies $y=(x - d)\tan\varphi$.

Step3: Equate the two expressions for $y$

Since $x\tan\theta=(x - d)\tan\varphi$, we can expand the right - hand side: $x\tan\theta=x\tan\varphi - d\tan\varphi$.

Step4: Solve for $x$

Rearrange the equation $x\tan\theta=x\tan\varphi - d\tan\varphi$ to get $x(\tan\theta-\tan\varphi)=-d\tan\varphi$, then $x = \frac{d\tan\varphi}{\tan\varphi-\tan\theta}$.

Step5: Substitute $x$ into the first $y$ equation

Substitute $x=\frac{d\tan\varphi}{\tan\varphi - \tan\theta}$ into $y = x\tan\theta$. We have $y=\frac{d\tan\theta\tan\varphi}{\tan\varphi-\tan\theta}$.

Answer:

$y=\frac{d\tan\theta\tan\varphi}{\tan\varphi - \tan\theta}$