How would you solve this Trigonometry Problem?

Question

While hiking on a level path toward Colorado's front range, Otis Evans determines that the angle of elevation to the top of Long's Peak is 30 degrees. Moving 1000 ft. closer to the mountain, Otis determines the angle of elevation to be 35 degrees. How much higher is the top of Long's Peak than Otis's elevation?

Answer 1

Let h be the height of Long's Peak. Let (x+1000) be Otis' initial distance from Long's Peak.

h = x * Tan(35π/180) = x * Tan(7π/36)

h = (x + 1000) * Tan(30π/180) = 1000 * Tan(π/6)

(x + 1000) * Tan(π/6) = x * Tan(7π/36)

Tan(π/6) * x + 1000 * Tan(π/6) = x * Tan(7π/36)

1000 * Tan(π/6) = x * (Tan(7π/36) - Tan(π/6))

x = 1000 * Tan(π/6) / (Tan(7π/36) - Tan(π/6))

x = 4699.36

h = 4699.36 * Tan(35*π/180) = 3290.53

Answer: 3290.53 ft

Answer 2

Let H be the height of Long's Peak above the plain, and let x be the distance from the 2nd observation point to the point directly below the peak.  Then we have the two equations to solve:

Tan(30) = H / (1000 + x)

Tan(35) = H / x

Doing a bit of basic algebra, we find that

H = (1000 Tan(30)Tan(35) / (Tan(35) - Tan(30)) = 3,290.53 approximately