Triangle Word Problem

Question

Two sets of wires are attached to the top of a tower to stabilize it. The closer wires are at a 50 degree angle to the ground. The second set is 40 feet further away from the first set and makes a 40 degree angle to the ground. How tall must the tower be?

Answer 1

You are going to use right triangle trig (sometimes also called SOHCAHTOA trig) to answer this.

Start by creating a diagram from the description. You will draw a smaller and bigger right triangle that share the vertical side which represents the tower. You have two unknowns, the height of the tower (h), and the distance from where the closer wire enters the ground and the base of the tower (call this x).

Notice in both triangles, you have info about the opposite and adjacent sides to the angles, so you will set up equations using tangent:

tan50 = h/x and tan40 = h/(x+40)

Solving for x in the first equation gives x = h/tan50. Substituting that expression for x in the 2nd equation :

tan40 = h/(h/tan50)+40) Now multiply both sides by the denominator.

htan40/tan50 + 40 tan40 = h Use a calculator to evaluate:

.704h + 33.564 = h And solve:

.296h = 33.564

h ~ 113.4 ft.

Answer 2

h is the height of the tower

x is the distance from the base of the tower to the 1st set of wires

the then distance from the base of the tower to the 2nd set of

wires is x+40

tan 50 = h/x

tan 40 = h/(x+40)

first equation says: x * tan 50 = h

plugs into second equation

tan 40 = (x * tan 50) / (x+40)

cross multiplies:

tan 40 (x+40) = x * tan 50

tan 40 x + tan 40* 40 = x * tan 50

tan 40*40 = x tan 50 - x tan 40

tan 40* 40 = x( tan 50 - tan 40)

tan 40 * 40 / ( tan 50 - tan 40) = x

x=95.175409662872670724328742185979

the height of the tower is just over 113