To solve this problem, we need to determine at what height each wire is connected to the mast and then find the distance between these two points on the ground.
### Step-by-Step Solution
1. Understanding the Problem:
- We have a mast held by two wires, both of which are 20 meters long.
- The first wire is attached to the ground 14 meters from the base of the mast.
- The second wire is attached to the ground 16 meters from the base of the mast.
2. Identify the Right Triangles:
- Each wire forms a right triangle with the mast and the ground.
- Let's name the triangles:
- Triangle 1: formed by the first wire, the mast, and a horizontal distance of 14 meters.
- Triangle 2: formed by the second wire, the mast, and a horizontal distance of 16 meters.
3. Use the Pythagorean Theorem:
- We apply the Pythagorean theorem to find the height at which each wire joins the mast. For a right triangle with sides 'a' (distance from the base), 'b' (height of the mast where the wire is attached), and hypotenuse 'c' (length of the wire), the theorem is:
[tex]\[
b^2 = c^2 - a^2
\][/tex]
4. Calculate the Heights:
- For Triangle 1:
[tex]\[
b_1 = \sqrt{20^2 - 14^2}
\][/tex]
[tex]\[
b_1 = \sqrt{400 - 196}
\][/tex]
[tex]\[
b_1 = \sqrt{204}
\][/tex]
[tex]\[
b_1 \approx 14.2828568570857 \text{ meters}
\][/tex]
- For Triangle 2:
[tex]\[
b_2 = \sqrt{20^2 - 16^2}
\][/tex]
[tex]\[
b_2 = \sqrt{400 - 256}
\][/tex]
[tex]\[
b_2 = \sqrt{144}
\][/tex]
[tex]\[
b_2 = 12.0 \text{ meters}
\][/tex]
5. Determine the Distance Between the Points on the Ground:
- The distance between these two points where the wires are attached to the ground is simply the absolute difference between the two horizontal distances.
[tex]\[
\text{Distance} = |16 - 14|
\][/tex]
[tex]\[
\text{Distance} = 2 \text{ meters}
\][/tex]
So the heights where the wires join the mast are approximately 14.2828568570857 meters and 12.0 meters respectively, and the distance between the points on the ground where the wires are attached is 2 meters.
### Final Answer:
The distance between the points where the wires join the mast is 2 meters.
