two wires support a pole in opposite directions. they make angles of 36° and 42° with the horizontal, and the points where they touch the ground are 50 ft apart. find the length of each wire
Draw a diagram.
If the pole has height h, then
h cot36° + h cot42° = 50
h = 20.1
Now, the wires have lengths
20.1/sin36° and 20.1/sin42°
To find the length of each wire, we can use trigonometry and create two right triangles. Let's consider the wire that makes an angle of 36° with the horizontal.
In the triangle formed by this wire, the angle 36° is opposite to the side connecting the top of the pole to the ground. We can label this side as h1. The side opposite to the right angle is known as the adjacent side, and its length is the distance between the point where the wire touches the ground and the base of the pole, which is 50 ft.
Now, we can use the trigonometric function tan to find the length of h1. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. So:
tan(36°) = h1 / 50
Rearranging the equation to solve for h1, we get:
h1 = 50 * tan(36°)
Now, let's consider the wire that makes an angle of 42° with the horizontal. Following the same steps as before, we can label the side connecting the top of the pole to the ground as h2. The adjacent side is still 50 ft.
Using the trigonometric function tan for this angle, we have:
tan(42°) = h2 / 50
Rearranging the equation for h2, we get:
h2 = 50 * tan(42°)
Therefore, the length of each wire is h1 and h2 respectively, which are equal to:
h1 = 50 * tan(36°)
h2 = 50 * tan(42°)