Kari wants to measure the height of a tree. She walks exactly 90 feet away from the base of the tree and looks up to the top of it. The angle from the ground to the top of the tree is 32 degrees. This particular tree grows at an angle of 85 degrees with respect to the ground rather than vertically. How tall is the tree to the nearest tenth of a foot? Enter a number answer only.

this is a Sin(), Cos() and Tan()  function  problem.   use the Mnemonic  (new-monic)  SOH CAH TOA  (soo caw tow ah ) to remember which function  sin() cos()  or tan() to use.. :) these types of problems are pretty confusing, Use that Mnemonic to keep the functions in order

Step-by-step explanation:

since we know the distance to the tree.. and we know the angle and we have a formula  TOA to figure out the one pieces of info we don't know ...  use TOA  that tells us our formula..  TOA(Ф)=Opp /  Adj     where  Tan(Ф) is using Ф= to our angle of 32 degrees ( remember Ф can be in degrees or radians, its in degrees for this problem) Adj = the Adjacent side of the triangle from the known angle.. the 32 degree angle ,  Opp = the Opposite side of the triangle formed by the imaginary lines from Kari to the base of the tree and from Kari to the top of the tree..   and then the tree , from it's base to it's top... we know the angle and the Adj side of that triangle so we can figure out that last part of TOA  by saying  Tan(Ф)=Opp/ Adj .... or say  Tan(32)= Opp / Adj    ⇒ Tan(32)=Opp / 90 ⇒ Tan(32) *90 = Opp   this will be the height from the ground to the top of the tree..  but.. the tree is at an angle , keep in mind... sooo  Tan(32)*90=56.238 feet  now take the new triangle that we know one side to... from the ground to the height of the tree..  in a vertical line from the ground to a point at 56.238 feet .. that is the Adj side of this new triangle.. the tree is on the Hypotenuse of this new trialge,  which we want to find. we know the Adj side we know the angle to find the Hypotenuse use CAH , from our Mnemonic at the top,  the Mnemonic is really important b/c keeping track of which part goes where in the triangles is pretty confusing.  Cos(5)=Adj / Hyp  ⇒ Cos(5)=56.238 / Hyp ⇒56.238 / Cos(5)= Hyp   = 56.453 feet  :)  that's how tall that tree is , lets chop it down :P and blame it on George Washington  :P  The tree, to the nearest 10th of a foot, is 56.5 feet