SOLUTION: A passenger on a boat at sea looks up to the top of a mountain, 350 m above sea level. The angle of elevation is 12°. After the boat sails towards the mountain, the angle of eleva

Question 1206872: A passenger on a boat at sea looks up to the top of a mountain, 350 m above sea level. The angle of elevation is 12°. After the boat sails towards the mountain, the angle of elevation is found to be 23°. How far did it sail towards the mountain?
Found 2 solutions by mananth, math_tutor2020:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!

A passenger on a boat at sea looks up to the top of a mountain, 350 m above sea level. The angle of elevation is 12°. After the boat sails towards the mountain, the angle of elevation is found to be 23°. How far did it sail towards the mountain?
tan 12 = 350/(x+y)
x+y = 350/tan12...................1
tan 23 = 350/x
x = 350/tan23...........................2
subtract 2 from 1
y = 350/tan12 - 350/tan 23
y=822.07 m
.

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answer: 822.072210279 meters approximately
Round this value however needed.

Explanation

Review this similar question
https://www.algebra.com/algebra/homework/Triangles/Triangles.faq.question.1199063.html
On that solution page I provide the derivation of this formula
h = d*u*v/(u-v)
where,
h = height or altitude of the object
d = distance between observation points
u = tangent of larger angle
v = tangent of smaller angle

We can easily solve for variable d.
h = d*u*v/(u-v)
h(u-v) = d*u*v
d = h*(u - v)/(u*v)

Then plug in the following information
h = 350
u = tan(23)
v = tan(12)
and we should get,

h = d*u*v/(u - v)
d = 350*(tan(23) - tan(12))/(tan(23)*tan(12))
d = 822.072210279 approximately
Please make sure that your calculator is set to degrees mode.

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