Chapter 8

At Wrigley Field in Chicago, the straight-line distance from home plate over second base to the center field wall is 400 feet. How far is it from first base to the same point at the enter field wall? IHint; Adapt and extend the figure .

A swimming pool is three feet deep in the shallow end. The bottom of the pool has a steady downward drop of \(12^{\circ} .\) If the pool is 50 feet long, how deep is it at the deep end?

Solve the triangle. The Law of Cosines may be needed. $$b=17.2, c=12.4, B=62.5^{\circ}$$

Solve the triangle. The Law of Cosines may be needed. $$b=24.1, c=10.5, C=26.3^{\circ}$$

Solve the triangle. The Law of Cosines may be needed. $$a=10.1, b=18.2, A=50.7^{\circ}$$

A buoy in the ocean is observed from the top of a 40 -meterhigh radar tower on shore. The angle of depression from the top of the tower to the base of the buoy is \(6.5^{\circ} .\) How far is the buoy from the base of the radar tower?

Solve the triangle. The Law of Cosines may be needed. $$b=14.6, c=7.8, B=40.4^{\circ}$$

One plane flies west from Cleveland at 350 mph. A second plane leaves Cleveland at the same time and flies southeast at 200 mph. How far apart are the planes after 1 hour and 36 minutes?

A 150 -foot-long ramp connects a ground-level parking lot with the entrance of a building. If the entrance is 8 feet above the ground, what angle does the ramp make with the ground?

Solve the triangle. The Law of Cosines may be needed. $$b=12.2, c=20, A=65^{\circ}$$

Solve the triangle. The Law of Cosines may be needed. $$a=44, c=84, C=42.2^{\circ}$$

Two ships leave port, one traveling in a straight course at \(22 \mathrm{mph}\) and the other traveling a straight course at \(31 \mathrm{mph}\) Their courses diverge by \(38^{\circ} .\) How far apart are they after 3 hours?

A man stands 20 feet from a statue. The angle of elevation from his eye level to the top of the statue is \(30^{\circ},\) and the angle of depression to the base of the statue is \(15^{\circ} .\) How tall is the statue?

A rocket shoots straight up from the launchpad. Five seconds after liftoff, an observer two miles away notes that the rocket's angle of elevation is \(3.5^{\circ} .\) Four seconds later, the angle of elevation is \(41^{\circ} .\) How far did the rocket rise during those four seconds?

Use the figure for Exercises 25–28. Solve the right triangle under the given conditions. $$a=8 \quad \text { and } \quad \Varangle A=40^{\circ}$$

From a 35 -meter-high window, the angle of depression to the top of a nearby streetlight is \(55^{\circ} .\) The angle of depression to the base of the streetlight is \(57.8^{\circ} .\) How high is the streetlight?

A plane takes off at an angle of \(6^{\circ}\) traveling at the rate of 200 feet/second. If it continues on this flight path at the same speed, how many minutes will it take to reach an altitude of 8000 feet?

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?

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. \(]\)

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?

Solve the triangle. The Law of Cosines may be needed. A straight path makes an angle of \(6^{\circ}\) with the horizontal. A statue at the higher end of the path casts a 6.5 -meter-long shadow straight down the path. The angle of elevation from the end of the shadow to the top of the statue is \(32^{\circ} .\) How tall is the statue?

One plane flies straight east at an altitude of 31,000 feet. A second plane is flying west at an altitude of 14,000 feet on a course that lies directly below that of the first plane and directly above the straight road from Thomasville to Johnsburg. As the first plane passes over Thomasville, the second is passing over Johnsburg. At that instant, both planes spot a beacon next to the road between Thomasville to Johnsburg. The angle of depression from the first plane to the beacon is \(61^{\circ},\) and the angle of depression from the second plane to the beacon is \(34^{\circ} .\) How far is Thomasville from Johnsburg?

A vertical statue 6.3 meters high stands on top of a hill. At a point on the side of the hill 35 meters from the statue's base, the angle between the hillside and a line from the top of the statue is \(10^{\circ} .\) What angle does the side of the hill make with the horizontal?

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.?

A fence post is located 50 feet from one corner of a building and 40 feet from the adjacent corner. Fences are put up between the post and the building corners to form a triangular garden area. The 40 -foot fence makes a \(58^{\circ}\) angle with the building. How long is the building wall?

A parallelogram has diagonals of lengths 12 and 15 inches that intersect at an angle of \(63.7^{\circ} .\) How long are the sides of the parallelogram? [See the hint for Exercise \(36 .]\)

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?

Two planes at the same altitude approach an airport. One plane is 16 miles from the control tower and the other is 22 miles from the tower. The angle determined by the planes and the tower, with the tower as vertex, is \(11^{\circ} .\) How far apart are the planes?

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)

A plane flies in a direction of \(105^{\circ}\) from airport \(A\). After a time, it turns and proceeds in a direction of \(267^{\circ} .\) Finally, it lands at airport \(B, 120\) miles directly south of airport \(A\) How far has the plane traveled? [ Note: Aerial navigation directions are explained in Exercise \(41 \text { of Section } 8.2 .]\)

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.

A 50 -foot-high flagpole stands on top of a building. From a point on the ground, the angle of elevation of the top of the pole is \(43^{\circ},\) and the angle of elevation of the bottom of the pole is \(40^{\circ} .\) How high is the building?

A plane flies in a direction of \(85^{\circ}\) from Chicago. It then turns and flies in the direction of \(200^{\circ}\) for 150 miles. It is then 195 miles from its starting point. How far did the plane fly in the direction of \(85^{\circ} ?\) (See the note in Exercise 46.)

Two points on level ground are 500 meters apart. The angles of elevation from these points to the top of a nearby hill are\(52^{\circ}\) and \(67^{\circ},\) respectively. The two points and the ground level point directly below the top of the hill lie on a straight line. How high is the hill?

Given triangle \(A B C,\) with \(B=60^{\circ}, a=7,\) and \(c=15,\) solve the triangle as follows. (a) Show that \(b=13 . \text { [Hint: Example } 1 \text { of Section } 8.3 .]\) (b) Use the Law of sines to find angle \(C\). (c) Use the fact that the sum of the angles is \(180^{\circ}\) to find angle \(A\)