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.