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?

Guest Jan 21, 2015

#1**+5 **

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.....

CPhill
Jan 22, 2015

#1**+5 **

Best Answer

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.....

CPhill
Jan 22, 2015