91Ó°ÊÓ

A visitor to the Leaning Tower of Pisa observed that the tower's shadow was 40 meters long and that the angle of elevation from the tip of the shadow to the top of the tower was \(57^{\circ} .\) The tower is now 54 meters tall (measured from the ground to the top along the center line of the tower). Approximate the angle \(\alpha\) that the center line of the tower makes with the vertical. (IMAGES CANNOT COPY)

A car on a straight road passes under a bridge. Two seconds later an observer on the bridge, 20 feet above the road, notes that the angle of depression to the car is \(7.4^{\circ} .\) How fast (in miles per hour) is the car traveling? [Note: 60 mph is equivalent to \(88 \text { feet/second. }]\)

A pole tilts at an angle \(9^{\circ}\) from the vertical, away from the sun, and casts a shadow 24 feet long. The angle of elevation from the end of the pole's shadow to the top of the pole is \(53^{\circ} .\) How long is the pole?

A plane passes directly over your head at an altitude of 500 feet. Two seconds later, you observe that its angle of elevation is \(42^{\circ} .\) How far did the plane travel during those two seconds?

One diagonal of a parallelogram is 6 centimeters long, and the other is 13 centimeters long. They form an angle of \(42^{\circ}\) with each other. How long are the sides of the parallelogram? [Hint: The diagonals of a parallelogram bisect each other. \(]\)

Assume that the earth is a sphere of radius 3960 miles. A satellite travels in a circular orbit around the earth, 900 miles above the equator, making one full orbit every 6 hours. If it passes directly over a tracking station at 2 P.M., what is the distance from the satellite to the tracking station at 2: 05 P.M.?

Each of two observers 400 feet apart measures the angle of elevation to the top of a tree that sits on the straight line between them. These angles are \(51^{\circ}\) and \(65^{\circ},\) respectively. How tall is the tree? How far is the base of its trunk from each observer?

A triangular piece of land has two sides that are 80 feet and 64 feet long, respectively. The 80 -foot side makes an angle of \(28^{\circ}\) with the third side. An advertising firm wants to know whether a 30 -foot long sign can be placed along the third side. What would you tell them?

From the top of the 800 -foot-tall Cartalk Tower, Tom sees a plane; the angle of elevation is \(67^{\circ} .\) At the same instant, Ray, who is on the ground, 1 mile from the building, notes that his angle of elevation to the plane is \(81^{\circ}\) and that his angle of elevation to the top of Cartalk Tower is \(8.6^{\circ} .\) Assuming that Tom and Ray and the airplane are in a plane perpendicular to the ground, how high is the airplane? (IMAGES CANNOT COPY)

Use the Law of Cosines to prove that the sum of the squares of the lengths of the two diagonals of a parallelogram equals the sum of the squares of the lengths of the four sides.