imagine you see a mountain in front of you. the angle of elevation to the peak is 3.8deg. when you drive 15mi. closer, the angle of elevation is 10deg. what is the height of the mountain? (nearest ft)

Answer 1

h is the height of the mountain.

tan 3.8 = h / (x+15)

tan 10 = h/x

x tan 10 = (x+15) tan 3.8

x tan 10 = x tan 3.8 + 15 tan 3.8

x = 24.06

this forces the height h = 1.598

Answer 2

This problem can be solved in two ways: using only right triangles, or using non-right angles, The second way requires more advance trig than the first way does, so I'll use the first approach. This way requires a dummy variable -- a variable used as a placeholder only and the value is not needed or evaluated. That dummy variable is the distance between the second stop and the point on the surface of the earth that is directly under the peak of the mountain, which will be designated as d.

Let

h = the height of the mountain.

d = the dummy variable referred to above

Now, the tangent of an angle in a right triangle is the ratio of opposite side to the adjacent side:

tan(3.8 deg) = h / (15 + d)

tan(10 deg) = h / d

From these two equations, we can solve for h. Start by taking the reciprocal of both sides of both equations:

cot(3.8 deg) = (15 + d) / h = 15 / h + d / h

cot(10 deg) = d / h

Subtracting:

cot(3.8 deg) - cot(10 deg) = 15 / h

Solving for h yields:

h = 15 / [cot(3.8 deg) - cot(10 deg) ]

h = 15 / [ 1/tan(3.8 deg) - 1/tan(10 deg) ]

h = 1.5984 miles = 8439.501 feet

Ans: 8440 ft

Answer 3

There are 2 equations in this problem. The first is when you are farther away from the mountain and one when you are closer.

Let y=height of the mountain

Let x= how far you are from the mountain once you drive 15 miles closer.

Since we have opposite and adjacent sides, you would use tangent.

tan 10 = y/x Multiply both sides by x

x tan 10 = y

tan 3.8 = y/(x+15) Multiply both sides by (x+15)

(x+15) tan 3.8 = y

Since we have both equations in terms of y, we can solve by substitution.

x tan 10 = (x+15) tan 3.8 Distribute to clear the parenthesis

x tan 10 = x tan 3.8 +15 tan 3.8 Subtract x tan 2.3 from both sides

x tan 10 - x tan 3.8 = x tan 2.3 +15 tan 3.8 - x tan 2.3 Simplify

x tan 10 - x tan 3.8 = 15 tan 3.8 Factor out x on the left

x(tan 10 - tan 3.8) = 15 tan 3.8 Divide both sides by (tan 10 - tan 3.8)

x(tan 10 - tan 3.8)/(tan 10 - tan 3.8) = 15 tan 3.8/(tan 10 - tan 3.8) Simplify

x = 15 tan 3.8/(tan 10 - tan 3.8) Make sure calculator is in degrees.

x=9.06 miles

You have to solve for y, that is the height of the mountain that the problem is asking for. Using our original equation.

x tan 10 = y

9.06 tan 10 = 1.59 miles x 5280 ft/mile = 8440 ft.