From a hand glider approaching a 6000 foot clearing the angles of depression of the opposite ends of the field measure 24 degrees and 30 degrees. how far is the hang glider from the nearer end of the field?

To find the distance of the hang glider from the nearer end of the field, we can use trigonometry and the concept of angles of depression.

Let's denote the distance from the hang glider to the nearer end of the field as 'x' (in feet).

Now, let's look at the geometry of the situation. The angles of depression represent the angles formed between the line of sight from the hang glider to the two ends of the field (on the ground) and the horizontal line (level ground). From the given information, we can visualize the situation as follows:

G (hang glider)
|\
| \
| \ x
|____\
A B C

Angle A = 24 degrees (angle of depression to the nearer end)
Angle C = 30 degrees (angle of depression to the farther end)
BC = 6000 feet (the length of the clearing)

We need to find the distance 'x' from the hang glider to the nearer end, AB.

Now, consider right-angled triangle ABG. Angle A in this triangle is equal to 180 - 24 = 156 degrees.

Using trigonometry, specifically the tangent function, we can relate the angle A and side lengths in the right-angled triangle ABG:

tan(A) = opposite/adjacent
tan(156) = x/BC
tan(156) = x/6000

Now, solving for 'x', we can rewrite the equation as follows:

x = 6000 * tan(156)

Plugging this into a calculator, we find:

x ≈ 23.47 feet

Therefore, the hang glider is approximately 23.47 feet from the nearer end of the field.