From a point on level ground, the angle of elevation to the top of a tower is 32°. After moving 50 meters closer to the tower, the angle of elevation is 47°. Find the height of the tower to the nearest tenth of a meter.

This problem creates two right triangles that share the same tower height.

Let x be the horizontal distance from the closer observation point to the base of the tower.

Because the original observation point is 50 meters farther away, its distance from the tower is x + 50.

Let h be the height of the tower.

From the closer point, use tangent:
tan(47°) = h / x

Solve for h:
h = x tan(47°)

From the farther point:
tan(32°) = h / (x + 50)

Solve for h:
h = (x + 50)tan(32°)

Both expressions equal the same tower height, so set them equal:
x tan(47°) = (x + 50)tan(32°)

Using calculator values:
tan(47°) ≈ 1.0724
tan(32°) ≈ 0.6249

Substitute:
1.0724x ≈ 0.6249(x + 50)

Distribute:
1.0724x ≈ 0.6249x + 31.245

Subtract 0.6249x from both sides:
0.4475x ≈ 31.245

Divide by 0.4475:
x ≈ 69.8

Now use the closer observation equation:
h = x tan(47°)

Substitute x ≈ 69.8:
h ≈ 69.8(1.0724)
h ≈ 74.9

Therefore, the tower is approximately 74.9 meters tall.