91Ó°ÊÓ

Solve for \(x\). Express each answer accurate to the nearest hundredth of a unit. $$\tan 29^{\circ}=\frac{x}{112}$$

Draw a regular decagon with side length \(6 \mathrm{cm}\) Divide the decagon into ten congruent isosceles triangles. a. Find the length of the apothem, and then use the Regular Polygon Area Conjecture to find the area of the decagon. b. Find the lengths of the legs of the isosceles triangles, and then use the SAS Triangle Area Conjecture to find the area of each isosceles triangle. Multiply to find the area of the decagon. c. Compare your answers from parts a and b.

Captain Malloy is flying a passenger jet. He is heading east at \(720 \mathrm{km} / \mathrm{h}\) when he sees an electrical storm straight ahead. He turns the jet \(20^{\circ}\) to the north to avoid the storm and continues in this direction for \(1 \mathrm{h}\). Then he makes a second turn, back toward his original flight path. Eighty minutes after his second turn, he makes a third turn and is back on course. By avoiding the storm, how much time did Captain Malloy lose from his original flight plan?

According to a Chinese legend from the Han dynasty (206 BCE-220 CE.), General Han Xin flew a kite over the palace of his enemy to determine the distance between his troops and the palace. If the general let out 800 meters of string and the kite was flying at a \(35^{\circ}\) angle of elevation, how far away was the palace from General Han Xin's position?

Benny is flying a kite directly over his friend Frank, who is 125 meters away. When he holds the kite string down to the ground, the string makes a \(39^{\circ}\) angle with the level ground. How high is Benny's kite? (IMAGE CAN'T COPY)

Archaeologists have recently started uncovering remains of James Fort (also known as Jamestown Fort) in Virginia. The fort was in the shape of an isosceles triangle. Unfortunately, one corner has disappeared into the James River. If the remaining complete wall measures 300 feet and the remaining corners measure \(46.5^{\circ}\) and \(87^{\circ},\) how long were the two incomplete walls? What was the approximate area of the original fort?

Application Dakota Davis uncovers the remains of a square-based Egyptian pyramid. The base is intact and measures 130 meters on each side. The top of the pyramid has eroded away, but what remains of each face of the pyramid forms a \(65^{\circ}\) angle with the ground. What was the original height of the pyramid?

A lighthouse 55 meters above sea level spots a distress signal from a sailboat. The angle of depression to the sailboat measures \(21^{\circ} .\) How far away is the sailboat from the base of the lighthouse?

A tree grows vertically on a hillside. The hill is at a \(16^{\circ}\) angle to the horizontal. The tree casts an 18 -meter shadow up the hill when the angle of elevation of the sun measures \(68^{\circ} .\) How tall is the tree?

Application A ship's officer sees a lighthouse at a \(42^{\circ}\) angle to the path of the ship. After the ship travels \(1800 \mathrm{m}\), the lighthouse is at a \(90^{\circ}\) angle to the ship's path. What is the distance between the ship and the lighthouse at this second sighting? (IMAGE CAN'T COPY)