Approximate the height of the cathedral. (Round your answer to the nearest whole number.)

Question

A cathedral is located on a hill. When the top of the spire is viewed from the base of the hill, the angle of elevation is 48°. When it is viewed at a distance of a = 100 feet from the base of the hill, the angle of elevation is 41°. The hill rises at an angle of 32°.

Answer 1

In the triangle including the base of the hill, 100 ft. from the base, and the top of the spire, we have the angles 41°, 180° - 48°, and 7°, and 100 ft. opposite the 7°, angle. We can use the sin law to get either of the other two sides. The side opposite the 41° angle will have length 100 * sin 41°/sin 7°. Then, the height of the spire is 100 * sin 41°/sin 7° * sin 48°. The base of the hill will be a horizontal distance of 100 * sin 41°/sin 7° * cos 48° from the church. Then, as the hill rises at 32°, the top of the hill will be at a height of tan 32° * 100 * sin 41°/sin 7° * cos 48°

Then, the height of the church is 100 * sin 41°/sin 7° * sin 48° - tan 32° * 100 * sin 41°/sin 7° * cos 48° =

400.057 - 225.086 = 174.970985170995 ft.