A tent pole is 85 feet high. A cable is attached to the ground 200 feet from the tower base. How long is the cable? What is the measure of the angle the cable makes to the ground?

The hypotenuse of a right triangle, using Pythagoras theorem, is

Length of cable
=√(200²+85²) feet

The angle would be
arc tangent(85/200)

To solve this problem, we can use the Pythagorean theorem and trigonometric functions. Let's break it down step by step:

Step 1: Draw a diagram
Draw a right-angled triangle to represent the situation. One side of the triangle represents the height of the tent pole (85 feet), the other side represents the distance from the tower base to the cable attachment point (200 feet), and the cable forms the hypotenuse.

|\
| \
| \
85 ft | \ Hypotenuse (cable)
| \
-------
200 ft

Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right-angled triangle. In this case, we have:
c^2 = a^2 + b^2

Substituting the values from our problem:
c^2 = 85^2 + 200^2

Step 3: Calculate the value of c
Calculate the square root of both sides to find the length of the cable (c):
c = sqrt(85^2 + 200^2)
c ≈ 217.36 feet

So, the length of the cable is approximately 217.36 feet.

Step 4: Calculate the angle the cable makes with the ground
To find the angle of the cable with the ground, we can use the inverse trigonometric function known as the arc tangent (tan^-1). We can use the tangent of the angle to find the ratio between the opposite side (85 feet) and the adjacent side (200 feet):

tan(angle) = opposite/adjacent
tan(angle) = 85/200

Step 5: Calculate the angle
Using the inverse tangent function (tan^-1) on both sides of the equation, we can find the angle:
angle = tan^-1(85/200)
angle ≈ 23.16 degrees

So, the cable makes an angle of approximately 23.16 degrees with the ground.