From one point on the ground, the angle of elevation to the top of a tree is measured at 36 degrees. From another point 20 feet closer, the angle of elevation is 65 degrees. How tall is the tree?
From the first observation, we have
tan 36 = h / x where h is the height of the tree and x is the distance the tree is from the first observation point
And solving for x, we have x = h / tan 36
And from the second observation, we have
tan 65 = h / (x - 20) and solving for x, we have
(x - 20) tan 65 = h
x - 20 = h /tan 65
x = h / tan 65 + 20
And setting the "x's" equal, we have
h / tan 36 = h/tan 65 + 20 rearrange
h / tan 36 - h / tan 65 = 20 multiply both sides by the common denominator, (tan 65 * tan 36)
h ( tan 65 - tan 36) = 20 (tan 65 * tan 36)
h = 20(tan (65) * tan (36) ) / (tan( 65) - tan (36)) = about 21.98 ft
Here's a pic.....
From the first observation, we have
tan 36 = h / x where h is the height of the tree and x is the distance the tree is from the first observation point
And solving for x, we have x = h / tan 36
And from the second observation, we have
tan 65 = h / (x - 20) and solving for x, we have
(x - 20) tan 65 = h
x - 20 = h /tan 65
x = h / tan 65 + 20
And setting the "x's" equal, we have
h / tan 36 = h/tan 65 + 20 rearrange
h / tan 36 - h / tan 65 = 20 multiply both sides by the common denominator, (tan 65 * tan 36)
h ( tan 65 - tan 36) = 20 (tan 65 * tan 36)
h = 20(tan (65) * tan (36) ) / (tan( 65) - tan (36)) = about 21.98 ft
Here's a pic.....