If Kate is flying a kite on the field and she lets out 20 meters of string and at the same time her friend Abigail is watching in the distance

The angle of elevation of the kite from her position is 65 degrees
And the angle of elevation of the kite from Abigail's position is 40 degrees

How far apart are they to the nearest meter

Let the distance be d.

If the kite is between the girls,

d/sin75° = 20/sin40°

But, if Kate is between the kite and Abigail, then

d/sin25° = 20/sin40°

To find the distance between Kate and Abigail, we can use trigonometry.

Let's assume AB represents the distance between Kate and Abigail, and let's assume AK represents the length of the string on the kite. In this case, AK is 20 meters.

We have two right-angled triangles: △AKE (with angle EKA = 65 degrees) and △ABE (with angle EBA = 40 degrees).

In triangle △AKE, we have the opposite side (AK) and the angle (65 degrees). We can use the tangent function to find the adjacent side (AE):
tan(65 degrees) = AE / AK

Similarly, in triangle △ABE, we have the opposite side (AK) and the angle (40 degrees). We can use the tangent function to find the adjacent side (BE):
tan(40 degrees) = BE / AK

Since AK is the same in both triangles, we can set the two expressions for AE and BE equal to each other:
AE / AK = BE / AK

Canceling out AK on both sides gives us:
AE = BE

This means that AE and BE are equal, so Kate and Abigail are the same distance from the kite, i.e., they are equidistant from the kite.

Therefore, the distance between Kate and Abigail is equal to AE or BE. To find this distance, we need to calculate AE.

Using the tangent function:
tan(65 degrees) = AE / AK
AE = AK * tan(65 degrees)
AE = 20 * tan(65 degrees)

Calculating this using a calculator, we find:
AE ≈ 35.905 meters

Therefore, Kate and Abigail are approximately 36 meters apart to the nearest meter.