A pole is braced by two wires that extended from the top of the pole to the ground. The lengths of the wires are 16 feet and 18 feet and the measure of the angle between the wires is 110 degrees. Find, to the nearest foot, the distance between the points at which the wires are fastened to the ground. Please show work !

using the law of cosines, the distance d is

d^2 = 16^2 + 18^2 - 2(16)(18)cos110°
d = 27.87 ft

To find the distance between the points at which the wires are fastened to the ground, we can use the Law of Cosines.

Let's assume that the distance between the points is represented by the variable 'd'.

Using the Law of Cosines, we can write:

d^2 = 16^2 + 18^2 - 2 * 16 * 18 * cos(110°)

Calculating:

d^2 = 256 + 324 - 2 * 16 * 18 * cos(110°)

d^2 = 580 - 576*cos(110°)

Now we can use a calculator to find the value of cos(110°):

cos(110°) ≈ -0.342

Substituting the value of cos(110°) into the equation:

d^2 ≈ 580 - 576 * (-0.342)

d^2 ≈ 580 + 197.952

d^2 ≈ 777.952

Taking the square root of both sides to solve for 'd':

d ≈ √777.952

d ≈ 27.9

Therefore, to the nearest foot, the distance between the points at which the wires are fastened to the ground is approximately 27.9 feet.

To find the distance between the points at which the wires are fastened to the ground, we can use the Law of Cosines.

The Law of Cosines states that in a triangle with sides a, b, and c and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, let's label the lengths of the wires as follows:
a = 16 feet
b = 18 feet
C = 110 degrees (angle between the wires)

We want to find the distance between the points at which the wires are fastened to the ground, which is represented by side c in the triangle.

Using the Law of Cosines, we can plug in the values:

c^2 = 16^2 + 18^2 - 2*16*18*cos(110)

c^2 = 256 + 324 - 576*cos(110)

Now, we calculate cos(110) using a calculator:

cos(110) ≈ -0.3420

Substituting this value back into the equation:

c^2 = 256 + 324 - 576*(-0.3420)

c^2 ≈ 256 + 324 + 197.664

c^2 ≈ 777.664

To find c, we take the square root of both sides:

c ≈ √(777.664)

c ≈ 27.891

Therefore, the distance between the points at which the wires are fastened to the ground is approximately 27.891 feet when rounded to the nearest foot.