Difficult Math Problem. Please Help?

Question

Two observers are 200 ft apart on opposite sides of a tree. The angles of elevation from the observers to the top of the tree are 30 degrees and 35 degrees. Find the height of the tree?

Please help and show work.

Answer 1

Draw the diagram

Let the height of the tree be h ft and the closer (ie the one with angle of elevation to tree of 35°) be x ft away from the tree
Thus the other is (200 - x) ft away

Then cot 35° = x/h
ie x = hcot 35°

Also cot 30° = (200 - x)/h
so 200 - x = hcot 30°
ie x = 200 - hcot 30°

Therefore hcot 35° = 200 - hcot 30°
ie h(cot35° + cot 30°) = 200
So h = 200/(cot 30° + cot 35°)
≈ 63.29 ft

Answer 2

Draw the figure. Label the distance from the 35 degree person to the tree as x, and the distance from the other person to the tree as 200 - x. Call the tree's height t.

On one side by SOHCAHTOA, tan 35 = t/x
Cross multiply to get t = x tan 35

On the other side, tan 30 = t/(200-x)


Cross multiply to get t = (200-x) tan 30

So since both equal t, x tan 35 = (200-x) tan 30

Do this out on your calculator to get x, then plug it into t = x tan 35 to get t.


NOW, if you are allowed to use Law of Sines, calculate the angle at the top of the tree (180 - 35 - 30 = 115) and use that to get one side of the big triangle:

Sin 115/200 = sin 30/y where y is the side attached to the 35 degree angle. Then once you get y, sin 35 = t/y. But you didn't say if you could use Law of Sines, or just right angle trig.

Answer 3

There are probably several approaches to this. i will show the area approach. The base of the triangle is 200'; the side opposite the 30º angle is found from the law of sines:

l = 200*sin(30º)/sin(180º-65º)

The area is given by the formula A = .5*l*200*sin(35) = .5*200^2*sin(30º)sin(35º)/sin(115º)

The area is also .5*h*200, where h = the height of the tree (altitude of the triangle), therefore

h = A/100

h = .5*(200^2/100)*sin(30º)sin(35º)/sin(115º)

h = 100*sin(30º)*sin(35º)/sin(115º)

Answer 4

let:
h = height of the tree
d = distance of the 30 degree person
200 - d = distance of the 35 degree person

tangent of 30 deg. = h / d
tangent of 35 deg. = h / (200 - d)

.5774 = h / d
.7002 = h / (200 - d)

solve for h and d:
d = 109.6 ft
h = 63.29 ft