Final answer:
Using trigonometric functions and the Pythagorean theorem, we solve for the height of the pole with the angle of elevation and the distances AB and AD, calculate the angles of elevation from points C and D, and determine the length of the rectangle's diagonal.
Step-by-step explanation:
To solve this trigonometry problem involving a vertical pole and a horizontal rectangular field, we can use basic trigonometric functions and the Pythagorean theorem.
- Calculation of the pole's height (a): We use the tangent function because we have the angle of elevation (22 degrees) and the distance from point A to the base of the pole (AB = 10M). Solving for the height (h) of the pole, tan(22°) = h/10. From this, we calculate h.
- Angle of elevation from point C (b): To find this, we'll need to determine the distance from point C to the base of the pole (PC) using the Pythagorean theorem for the rectangle ABCD, and then use the arc tangent function with the height of the pole and the distance PC.
- Length of rectangle diagonal (c): Again using the Pythagorean theorem, we compute the diagonal AC by finding the square root of the sum of the squares of AB and AD.
- Angle of elevation from point D (d): To find this angle, we can use the tangent function with the height of the pole and the distance from point D to the base of the pole (PD = AD).
By processing these steps, we can accurately determine the height of the pole, the required angles of elevation, and the diagonal length of the rectangle.