91Ó°ÊÓ

HEIGHT A flagpole at a right angle to the horizontal is located on a slope that makes an angle of \(12^{\circ}\) with the horizontal. The flagpole's shadow is 16 meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is \(20^{\circ}\). (a) Draw a triangle to represent the situation. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation that can be used to find the height of the flagpole. (c) Find the height of the flagpole.

DISTANCE Two ships leave a port at 9A.M. One travels at a bearing of N \(54^{\circ}\)W at 12 miles per hour, and the other travels at a bearing of S \(67^{\circ}\)W at 16 miles per hour. Approximate how far apart they are at noon that day.

BRIDGE DESIGN A bridge is to be built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is S \(41^{\circ}\)W. From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are S \(74^{\circ}\)E and S \(28^{\circ}\)E, respectively. Find the distance from the gazebo to the dock.

RAILROAD TRACK DESIGN The circular arc of a railroad curve has a chord of length 3000 feet corresponding to a central angle of \(40^{\circ}\). (a) Draw a diagram that visually represents the situation.Show the known quantities on the diagram and use the variables \(r\) and \(s\) to represent the radius of the arc and the length of the arc, respectively. (b) Find the radius \(r\) of the circular arc. (c) Find the length \(s\) of the circular arc.

GLIDE PATH A pilot has just started on the glide path for landing at an airport with a runway of length 9000 feet. The angles of depression from the plane to the ends of the runway are \(17.5^{\circ}\) and \(18.8^{\circ}\). (a) Draw a diagram that visually represents the situation. (b) Find the air distance the plane must travel until touching down on the near end of the runway. (c) Find the ground distance the plane must travel until touching down. (d) Find the altitude of the plane when the pilot begins the descent.

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle 3, 15 \rangle\) \(\mathbf{v} = \langle -1, 5 \rangle\)

DISTANCE A family is traveling due west on a road that passes a famous landmark. At a given time the bearing to the landmark is N \(62^{\circ}\)W, and after the family travels 5 miles farther the bearing is N \(38^{\circ}\)W. What is the closest the family will come to the landmark while on the road?

ALTITUDE The angles of elevation to an airplane from two points \(A\) and \(B\) on level ground are \(55^{\circ}\) and \(72^{\circ}\) respectively. The points \(A\) and \(B\) are 2.2 miles apart, and the airplane is east of both points in the same vertical plane. Find the altitude of the plane.

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle \cos\ \theta\), \(\sin\ \theta \rangle\) \(\mathbf{v} = \langle \sin\ \theta\), \(-\cos\ \theta \rangle\)

REVENUE The vector \(\mathbf{u} = \langle 3140, 2750 \rangle\) gives the numbers of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month. The vector \(\mathbf{v} = \langle 2.25, 1.75 \rangle\) gives the prices (in dollars) of the food items. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and interpret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 2.5%.