Question

While visiting Yosemite National Forrest, Joe approximated the angle of elevation to the top of a hill to be 25 degrees. After walking 350 ft closer, he guessed that the angle of elevation had increased by 14 degrees. Approximately how tall is the hill?

Answer 1

Answer: 384.780390296928 feet

This value is approximate. Round it however you need to.

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Work Shown:

Refer to the attached image

Point A = initial observation point
Point B = new observation point
Point C = bottom of the hill (directly underneath point D)
Point D = top of the hill (directly overhead point C)

Segment lengths
AB = 350
BC = x
CD = y
AC = AB+BC = 350+x

The goal is to find the value of y.

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tan(angle) = opposite/adjacent
tan(25) = CD/AB
tan(25) = y/(350+x)
(350+x)*tan(25) = y
y = (350+x)*tan(25)
y = (350+x)*0.466307658155
y = 350*0.466307658155+x*0.466307658155
y = 163.20768035425+0.466307658155x
y = 0.466307658155x+163.20768035425

tan(angle) = opposite/adjacent
tan(39) = CD/BC
tan(39) = y/x
tan(39) = (0.466307658155x+163.20768035425)/x
x*tan(39) = 0.466307658155x+163.20768035425
x*0.809784033195 = 0.466307658155x+163.20768035425
0.809784033195x-0.466307658155x = 163.20768035425
0.34347637504x = 163.20768035425
0.34347637504x/0.34347637504 = 163.20768035425/0.34347637504
x = 475.164209868127

Now use this x value to find y
y = 0.466307658155x+163.20768035425
y = 0.466307658155*475.164209868127+163.20768035425
y = 384.780390296928

Therefore, the hill is approximately 384.780390296928 feet tall.