Radius of the Earth A satellite is circling 112 miles

Geometry The two equal sides of an isosceles triangle are each 42 centimeters. If the base measures 32 centimeters, find the height and the measure of the two equal angles.

Geometry An equilateral triangle (one with all sides the same length) has an altitude of 4.3 inches. Find the length of the sides.

Geometry The height of a right circular cone is 25.3 centimeters. Ifthe diameter of the base is 10.4 centimeters, what angle does the side of the cone make with the base (Figure 12)?

Geometry The diagonal of a rectangle is 348 millimeters, while the longer side is 278 millimeters. Find the shorter side of the rectangle and the angles the diagonal makes with the sides.

Length of an Escalator How long should an escalator be if it is to make an angle of 33 with the floor and carry people a vertical distance of 21 feet between floors?

Height of a Hill A road up a hill makes an angle of 5.10 with the horizontal. If the road from the bottom of the hill to the top of the hill is 2.5 miles long, how high is the hill?

Length of a Rope A 72.5-foot rope from the top of a circus tent pole is anchored to the ground 43.2 feet from the bottom of the pole. What angle does the rope make with the pole? (Assume the pole is perpendicular to the ground.)

Angle of a Ladder A ladder is leaning against the top of a 7.0-foot wall If the bottom of the ladder is 4.5 feet from the wall, what is the angle between the ladder and the wall?

Angle of Elevation If a 73.0-foot flagpole casts a shadow 51.0 feet long, what is the angle of elevation of the sun (to the nearest tenth of a degree)?

Angle of Elevation Ifthe angle of elevation of the sun is 63.4 when a building casts a shadow of 37.5 feet, what is the height of the building?

Angle of Depression A person standing 150 centimeters from a mirror notices that the angle of depression from his eyes to the bottom of the mirror is 12, while the angle of elevation to the top of the mirror is 110. Find the vertical dimension of the mirror (Figure 13). Figure 13

Width of a Sand Pile A person standing on top of a 15-foot high sand pile wishes to estimate the width of the pile. He visually locates two rocks on the ground below at the base of the sand pile. The rocks are on opposite sides of the sand pile, and he and the two rocks are in the same vertical plane. If the angles of depression from the top ofthe sand pile to each of the rocks are 27 and 19, how far apart are the rocks?

Figure 14 shows the topographic map we used in Example 4 of this section. Recall that Stacey is at position S and Amy is at position A. In Figure 14, Travis, a third hiker, is at position T.Topographic Map Reading If the distance between A and T on the map in Figure 14 is 0.50 inch, find each of the following: a. the horizontal distance between Amy and Travis b. the difference in elevation between Amy and Travis c. the angle of elevation from Travis to Amy

Figure 14 shows the topographic map we used in Example 4 of this section. Recall that Stacey is at position S and Amy is at position A. In Figure 14, Travis, a third hiker, is at position T.Topogra~hic Map Reading If the distance between S and T on the map in Fig 14. ure 14 is "8 inch, find each of the following: a. the horizontal distance between Stacey and Travis h. the difference in elevation between Stacey and Travis c. the angle of elevation from Travis to Stacey

Figure 14 shows the topographic map we used in Example 4 of this section. Recall that Stacey is at position S and Amy is at position A. In Figure 14, Travis, a third hiker, is at position T.Height of a Door From a point on the floor the angle of elevation to the top of a door is 47, while the angle of elevation to the ceiling above the door is 59. If the ceiling is 9.8 feet above the floor, what is the vertical dimension of the door (Figure 15)? A II I I I I I I I I I I I I I I I I I I / I Figure 15

Figure 14 shows the topographic map we used in Example 4 of this section. Recall that Stacey is at position S and Amy is at position A. In Figure 14, Travis, a third hiker, is at position T.Height of a BOOding A man standing on the roof of a building 60.0 feet high looks down to the building next door. He finds the angle of depression to the roof of that building from the roof of his building to be 34.5, while the angle of depression from the roof of his building to the bottom of the building next door is 63.2. How taJI is the building next door?

Distance and Bearing Problems 17 through 22 involve directions in the form of bearing, which we defined in this section. Remember that bearing is always measured from a north-south line.A boat leaves the harbor entrance and travels 25 miles in the direction N 420 E. The captain then turns the boat 900 and travels another 18 miles in the direction S 480 E. At that time, how far is the boat from the harbor entrance, and what is the bearing of the boat from the harbor entrance (Figure 16)?

Distance and Bearing Problems 17 through 22 involve directions in the form of bearing, which we defined in this section. Remember that bearing is always measured from a north-south line.A man wandering in the desert walks 2.3 miles in the direction S 31 W. He then turns 900 and walks 3.5 miles in the direction N 59 W. At that time, how far is he from his starting point, and what is his bearing from his starting point?

Distance and Bearing Problems 17 through 22 involve directions in the form of bearing, which we defined in this section. Remember that bearing is always measured from a north-south line.Lompoc, California, is 18 miles due south of Nipomo. Buellton, California, is due east of Lompoc and S 65 E from Nipomo. How far is Lompoc from Buellton?

Distance and Bearing Problems 17 through 22 involve directions in the form of bearing, which we defined in this section. Remember that bearing is always measured from a north-south line.A tree on one side of a river is due west of a rock on the other side of the river. From a stake 21.0 yards north of the rock, the bearing of the tree is S 18.2 W. How far is it from the rock to the tree?

Distance and Bearing Problems 17 through 22 involve directions in the form of bearing, which we defined in this section. Remember that bearing is always measured from a north-south line.A boat travels on a course of bearing N 37 10' W for 79.5 miles. How many miles north and how many miles west has the boat traveled?

Distance and Bearing Problems 17 through 22 involve directions in the form of bearing, which we defined in this section. Remember that bearing is always measured from a north-south line.A boat travels on a course of bearing S 63 50' E for 101 miles. How many miles south and how many miles east has the boat traveled?

Distance In Figure 17, a person standing at point A notices that the angle of elevation to the top of the antenna is 47 30'. A second person standing 33.0 feet farther from the antenna than the person at A finds the angle of elevation to the top ofthe antenna to be 42 10'. How far is the person atA from the base of the antenna?

Height of an Obelisk Two people decide to find the height of an obelisk. They position themselves 25 feet apart in line with, and on the same side of, the obelisk. Ifthey find that the angles of elevation from the ground where they are standing to the top ofthe obelisk are 65 and 44, how tall is the obelisk? Figure 17 Figure 18

Qeight of a Tree An ecologist wishes to find the height of a redwood tree that is on the other side of a creek, as shown in Figure 18. From point A he finds that the angle of elevation to the top of the tree is 10.7. He then walks 24.8 feet at a right angle from point A to pointB. There he finds that the angle betweenAB and a line extending from B to the tree is 86.6. What is the height of the tree?

Rescue A helicopter makes a forced landing at sea. The last radio signal received nea at station C gives the bearing of the helicopter from Cas N 57.5 E at an altitude of 426 feet. An observer at C sights the helicopter and gives LDCB as 12.3. How ~O E. far will a rescue boat at A have to travel to reach any survivors at B (Figure 19)? :tion s the then ar is mia, 'rom iver. oW. Lany liles eva Figure 19 Figure 20

Height of a Flagpole Two people decide to estimate the height of a flagpole. 'hey One person positions himself due north of the pole and the other person stands the due east of the pole. If the two people are the same distance from the pole and are 25 feet from each other, find the height of the pole if the angle of elevation from the ground to the top of the pole at each person's position is 56 (Figure 20).

Height of a Tree To estimate the height of a tree, one person positions himself due south of the tree, while another person stands due east of the tree. If the two people are the same distance from the tree and 35 feet from each other, what is the height of the tree if the angle of elevation from the ground at each person's position to the top of the tree is 48?

Radius of the Earth A satellite is circling 112 miles above the earth, as shown in Figure 21. When the satellite is directly above point B, angle A is found to be 76.6. Use this information to find the radius ofthe earth.

Distance Suppose Figure 21 is an exaggerated diagram of a plane fiying above the earth. If the plane is 4.55 miles above the earth and the radius of the earth is 3,960 miles, how far is it from the plane to the horizon? What is the measure of angle A?

Distance A ship is anchored off a long straight shoreline that runs north and south. From two observation points 15 miles apart on shore, the bearings of the ship are N 31 0 E and S 53 E. What is the shortest distance from the ship to the shore?

Distance Pat and Tim position themselves 2.5 miles apart to watch a missile hat launch from Vandenberg Air Force Base. When the missile is launched, Pat estiita mates its bearing from him to be S 75 W, while Tim estimates the bearing of the md missile from his position to be N 65 W. IfTim is due south of Pat, how far is Tim from the missile when it is launched?

Spiral of Roots Figure 22 shows the Spiral of Roots we mentioned in the previous chapter. Notice that we have labeled the angles at the center of the spiral with fh, O2, 03, and so on. Figure 22Find the values of OJ, O2, and 03, accurate to the nearest hundredth of a degree.

Spiral of Roots Figure 22 shows the Spiral of Roots we mentioned in the previous chapter. Notice that we have labeled the angles at the center of the spiral with fh, O2, 03, and so on. Figure 22If 0" stands for the nth angle formed at the center of the Spiral of Roots, find a formula for sin 0".

The following problems review material we covered in Section 1.5.Expand and simplify: (sin 0 cos oi

The following problems review material we covered in Section 1.5.Subtract: -- cos 0

Show that each of the following statements is true by transforming the left side of each one into the right side.sin 0 cot 0 = cos 0

Show that each of the following statements is true by transforming the left side of each one into the right side.cos 0 csc 0 tan 0

Show that each of the following statements is true by transforming the left side of each one into the right side.sec 0 = csc 0

Show that each of the following statements is true by transforming the left side of each one into the right side.(1 cos 0)(1 + cos 0) = sin2 0

Show that each of the following statements is true by transforming the left side of each one into the right side.sec 0 - cos ()

Show that each of the following statements is true by transforming the left side of each one into the right side.1 -- = sin2 ()

One of the items we discussed in this section was topographic maps. The process of making one of these maps is an interesting one. It involves aerial photography and different colored projections of the resulting photographs. Research the process used to draw the contour lines on a topographic map, and then give a detailed explanation of that process.

Albert lives in New Orleans. At noon on a summer day, the angle of elevation of the sun is 84. The window in Albert's room is 4.0 feet high and 6.5 feet wide. (See Figure 23.) a. Calculate the area of the floor surface in Albert's room that is illuminated by the sun when the angle of elevation of the sun is 840 b. One winter day, the angle of elevation of the sun outside Albert's window is 37. Will the illuminated area of the floor in Albert's room be greater on the summer day, or on the winter day?