Answer
The height of the top of Mount Whitney above the level road is 2.5 km
Work Step by Step
Let $x$ be the distance from the vertical line to the point with an elevation of $22.667^{\circ}$.
We can write an expression for the height $h$:
$\frac{h}{x} = tan~22.667^{\circ}$
$h = x~tan~22.667^{\circ}$
We can use the second point to write another equation for the height $h$:
$\frac{h}{x+7} = tan~10.833^{\circ}$
$h = (x+7)~tan~10.833^{\circ}$
We can equate the two expressions to find $x$:
$x~tan~22.667^{\circ} = (x+7)~(tan~10.833^{\circ})$
$0.4176~x = 0.1914~x+1.3395$
$0.4176~x - 0.1914~x = 1.3395$
$x = \frac{1.3395}{0.2262}$
$x = 5.9218~km$
We can use the first equation to find $h$:
$h = x~tan~22.667^{\circ}$
$h = (5.9218~km)~tan~22.667^{\circ}$
$h = 2.5~km$
The height of the top of Mount Whitney above the level road is 2.5 km
