what is the lenght of the pole?

Question

a pole cast a shadow of 15 meter long when the angle of elevation of the sun is 61 degree. If the pole has leaned 15 degree from the vertical directly toward the sun, what is the lenght of the pole?

Answer 1

Draw a diagram showing the ground, leaning pole, perpendicular line from the top of the pole to the ground, and line from the top of the pole to where the shadow touches the ground.

X = Length of the pole

Y = Distance from the pole top to the ground

Z = Distance from pole bottom to point on the ground directly under the pole top

Label the known angles and the length of the shadow. Then,

Y = X (cos15) = 0.966X, and Z = X (sin15) = 0.2588X

Using the larger triangle,

tan61 = Y / (Z + 15 meters) = 0.966X / (0.2588X + 15)

Solve for X.

For practice, see if you can determine the length of the pole if it were leaning AWAY from the sun. (18.9 meters)

Answer 2

Draw a diagram. Let AB be a line from the top of the pole (A) to the ground (B) such that the angle B is 61 degrees. Let BC be a line from B to the pole base C. BC is 15 meter (the shadow). We want to find AC.
If the pole was vertical, angle A would be 29 degrees. Since it is tilted 15 degrees, the angle is 44. The law of sines can be used to find AC
Sin 44/15 = Sin 61/AC