What angle must the 14-foot-long wire make with the ground when a pole is supported by two wires of different lengths?

To find the angle that the 14-foot wire makes with the ground, we can use some basic trigonometry. We know the lengths of the wires and the distance between their anchor points on the ground.

We have:

• The length of the first wire (14 feet)
• The length of the second wire (17 feet)
• The distance between the points where the wires are secured to the ground (22 feet)

We can visualize this as a triangle where:

• The first wire is one side (14 feet).
• The second wire is another side (17 feet).
• The distance between the two attachment points (22 feet) is the base.

Using the Law of Cosines, we can find the angle the 14-foot wire makes with the ground:

Let A be the angle we want to find, and the three sides of our triangle be:

• Side a = 17 feet (second wire)
• Side b = 14 feet (first wire)
• Side c = 22 feet (distance between anchors)

According to the Law of Cosines:

c² = a² + b² - 2ab * cos(A)

Substituting the values we have:

22² = 17² + 14² - 2 * 17 * 14 * cos(A)

Calculating:

• 484 = 289 + 196 – 476 * cos(A)
• 484 = 485 – 476 * cos(A)
• 476 * cos(A) = 1
• cos(A) = 1/476

Now, we can find the angle A by taking the cosine inverse:

A = cos⁻¹(1/476)

Using a calculator, we find:

A ≈ 89.87 degrees

So, the angle that the 14-foot wire makes with the ground is approximately 89.87 degrees. This means it is almost vertical, due to the nature of the lengths of the wires and the distance apart.