I have two questions that I need help with.

1. Angle of Depression: A Global Positioning System satellite orbits 12,500 miles above Earth's surface. Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 4000 miles.

2. Geometry: A regular hexagon (a hexagon with congruent sides and angles) is inscribed in a circle of radius 25 inches. Find the length of the sides of the hexagon.

1. Measured from the vertical direction to the center of the Earth, the horizon is at angle sin^-1 (4000/16,500)

2. The side length of a hexagon is the radius of the circle in which it is inscribed.

Satellites in the global positioning system (gps) orbit is 12,500 miles above Earth. Gps signals can't travel through earth so a satellite at point b can transmit signals only to points on ac. How far must the satellites be able to transmit to reach points a and c. The diameter of earthy is about 8000

Sure! I can help you with both of these questions. Let's take them one by one.

1. Angle of Depression:
To find the angle of depression, we can use trigonometric ratios. In this scenario, we have a right triangle formed by the satellite, Earth's surface, and the line of sight to the horizon. The angle of depression is the angle between the line of sight and the horizontal.

To solve this, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the satellite (12,500 miles) minus the radius of the Earth (4000 miles), and the adjacent side is the radius of the Earth (4000 miles).

So, the tangent of the angle of depression can be written as:

tan(angle) = (height of satellite - radius of Earth) / radius of Earth

Plugging in the values, we get:

tan(angle) = (12,500 - 4000) / 4000
tan(angle) = 8500 / 4000

Using a calculator or trigonometric tables, we can find the angle whose tangent is 8500/4000. The inverse tangent (arctan) of this value will give us the angle.

2. Geometry:
To find the length of the sides of the hexagon inscribed in a circle, we can use the properties of a regular hexagon. A regular hexagon consists of six congruent equilateral triangles.

In an equilateral triangle, all sides are equal, and each angle measures 60 degrees. In the hexagon, there are six angles of 60 degrees.

Let's consider one of the equilateral triangles inscribed in the circle. The radius of the circle is given as 25 inches. In an equilateral triangle, the radius is also the height of each triangle.

Using the properties of the triangle, we can use the sine function to find the length of each side.

sin(angle) = opposite / hypotenuse
sin(60) = side length / radius

Plugging in the values, we get:

sin(60) = side length / 25
√3/2 = side length / 25

Multiplying both sides by 25, we get:

side length = 25 * √3/2
side length = 12.5 * √3

So the length of each side of the hexagon is 12.5 * √3 inches.

I hope this helps! Let me know if you have any further questions.

*Fart Noise