The angle of elevation of the top of a flagpole is 64° from a point 32m away from the foot of the flagpole. Find the angle of elevation of a flag halfway up the flagpole from that point correct to the nearest degree.

Answer 1

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.

Answer 2

Let A be the point where the observer is, C be the foot of the flagpole, B be the top of the flagpole, and D be a point on CB such that AD is perpendicular to DC.

Triangle ADC is a right triangle since angle ADC is 90°. The angle of elevation of point B (top of flagpole) is 64°. Thus, angle CAD = 180 - 90 - 64 = 26°.

Now, we are given that AC = 32m. To find AD, we use the sine function:

sin(CAD) = opposite / hypotenuse
sin(26°) = AD / 32
AD = 32 * sin(26°) = 32 * 0.4384 ≈ 14.03m

Since point D is halfway up the flagpole, DC is also equal to 14.03m.

In right triangle ADC, we have AD and DC equal to 14.03m each. Thus, triangle ADC is an isosceles right triangle, which means angle DCA = 45°.

Now let E be a point on CB such that AE is perpendicular to BE.

Triangle ABE is also right triangle with angle AEB = 90°. Furthermore, angle BAE is equal to angle DCA, since they are both equal to the angle of elevation of a flag halfway up the flagpole from point A. Thus, angle AEB is also 45°, which makes triangle ABE an isosceles right triangle as well.

Since triangle ABE is isosceles, AE = EB. To find AE, we can use the sine function again:

sin(BAE) = opposite / hypotenuse
sin(45°) = AE / (14.03m)
AE = 14.03m * sin(45°) = 14.03m * 0.7071 ≈ 9.92m

Thus, the angle of elevation of a flag halfway up the flagpole from that point is approximately 45°.