1. A surveyor can measure the width of a river by setting up a transit at a point C on one side of the river and taking a sighting of a point A on the other side. After turning through an angle of 90° at C, the surveyor walks 200 meters to point B. Using the transit at B, the angle is measured and found to be 20°. What is the width of the river rounded to the nearest meter? 2. An observer measures an angle of elevation of 41° for the top of a lighthouse with respect to a certain point on the ground. She then walks 15 feet closer to the lighthouse and measures an angle of elevation of 44°. How tall is the lighthouse? 1. Two surveyors need to find the distance across a lake. They place a reference pole at point A in the diagram. Point B is 3 m. east and 1 m. north of reference point A. Point C is 19 m. east and 13 m. north of point A. Find the distance across the lake, from B to C. 2. Determine the equation of the line passing through the points (-5, 3) and (4,2). 3. Find the equation of the line perpendicular x 4y = 3 and passing through the point (2,-5). 4. Determine the distance from the point (5, -3) to the line 2x - 4y + 10 = 0 3. A tree growing on hillside casts 10.2 meters shadow straight downhill. Find the vertical height of the tree if, relative to the horizontal, the hill slopes 15° and the angle of elevation of the sun is 62° 5. Determine the acute angle between the lines 4x - 3v + 9 = 0 and 3x - 8y + 1 = 0 4. From a building across, the angle of depression of the base of the front edifice is 23.5° and the angle of elevation to the top is 52.78°. The height of observation is 14 m. What is the height of the edifice? 7. An arch 18 m high has the form of parabola with a vertical axis. The length of a horizontal beam placed across the arch 8 m from the top is 64m. Find the width of the arch at the bottom. 5. Triangle ABC has the following data: b = 19 and A = 50 deg How many triangles may be formed if a = 12? a = 17? a = 23? 6. A tour boat travels 10 miles due north from its starting point and then travels in the direction S40°W on its second leg of the tour. If the boat is currently 15 miles from its starting point, how far has it traveled on the second leg? 7. Two motorboats start from the same point. One of them travels in the direction N30°W at 45 miles per hour, and the other heads due east at 35 miles per hour. How far apart are the boats after 1 hour? 8. A ship sails 15.0 mi on a course S40°10'W and then 21.0 mi on a course N28°20'W. Find the distance and direction of the last position from the first. 6. Determine the center of the circle with equation 𝑥 2 + 𝑦 2 − 4𝑥 + 6𝑦 − 23 = 0 8. A mirror for a reflecting telescope has the shape of a (finite) paraboloid of diameter 8 inches and depth 1 inch. How far from the center of the mirror will the incoming light collect? 9. The arch of a bridge is in the shape of a semiellipse having a horizontal span of 90 m and a height of 30 m at its center. How high is the arch 25 m to the right or left of the center? 10. Assume that the length of the major axis of Earth's orbit is 186,000,000 miles and that the eccentricity is 0.017. Approximate, to the nearest 1000 miles, the maximum and minimum distances between Earth and the sun. 11. A cooling tower is a hyperbolic structure. Suppose its base diameter is 100 meters and its smallest diameter of 48 meters occurs 84 meters from the base. If the tower is 120 meters high, approximate its diameter at the top.
