Let the height of the flagpole be h. Then we can use the tangent function to find h:
tan(64°) = h/32
h = 32 tan(64°)
h ≈ 74.9 m
Now, let x be the distance from the point to the midpoint of the flagpole. Then we can use the tangent function again to find the angle of elevation of the flag halfway up the flagpole:
tan(θ) = (h/2) / x
θ = tan⁻¹[(h/2) / x]
θ = tan⁻¹[(74.9/2) / x]
θ ≈ tan⁻¹[37.45 / x]
We don't know x, but we can use the fact that the midpoint of the flagpole is halfway between the foot of the flagpole and the top of the flagpole. Therefore, the distance from the midpoint of the flagpole to the foot of the flagpole is 74.9/2 = 37.45 m. Using the Pythagorean theorem, we have:
(32m)² + (37.45m)² = x²
x ≈ 48.5 m
Substituting into our formula for θ, we have:
θ ≈ tan⁻¹[37.45 / (48.5)]
θ ≈ tan⁻¹[0.771]
Using a calculator, we find:
θ ≈ 36°
Therefore, the angle of elevation of the flag halfway up the flagpole is approximately 36 degrees, to the nearest degree.