EXAMPLES.

In working these examples use logarithms or not, as appears most convenient. Check the results.

1. A ladder 28 ft. long is leaning against the side of a house, and makes an angle 27° with the wall. Find its projections upon the wall and upon the ground.

2. What is the projection of a line 87 in. long upon a line inclined to it at an angle 47° 30' ?

3. What are the projections: (a) of a line 10 in. long upon a line inclined 22° 30' to it? (b) of a line 27 ft. 6 in. long upon a line inclined 37° to it? (c) of a line 43 ft. 7 in. long upon a line inclined 67° 20' to it? (d) of a line 34 ft. 4 in. long upon a line inclined 55° 47' to it?

There are various

29. Measurement of heights and distances. instruments used for measuring angles. The sextant can be used for measuring the angle between the two lines drawn from the observer's eye to each of two distant objects. Horizontal and vertical angles can be measured with a theodolite or engineer's transit. When great accuracy is not required, vertical angles can be measured by means of a quadrant.

When an object is above the observer's eye, the angle between the line from the eye to the object, and the horizontal line through the eye and in the same vertical plane as the first line, is called the angle of elevation of the object, or simply the elevation of the object. When the object is below the observer's eye, this angle is called the angle of depression of the object, or simply the depression of the object.

Angle of Depression

Angle of Elevation

FIG. 21.

NOTE. In ordinary work engineers get angular measurements exact to within one minute, and in the best ordinary work to half a minute. In very particular work, like geodetic survey, they can get measurements exact to five seconds. For ordinary engineering work five-place tables are generally used; four-place tables are used in some kinds of work. See Art. 11. Note 1, Art. 27, Ex. 2, Note.

EXAMPLES.

A few simple examples are given here; others will be given later.

1. At a point 150 ft. from, and on a level with, the base of a tower, the angle of elevation of the top of the tower is observed to be 60°. Find the height of the tower.

Let AB be the tower, and P the point of observation.

B

AB=AP tan 60°=150× √3=150×1.7321=259.82 ft.

FIG. 22.

2. In order to find the height of a hill, a line was measured equal to 100 ft., in the same level with the base of the hill, and in the same vertical plane with its top. At the ends of this line the angles of elevation of the top of the hill were 30° and 45°. Find the height of the hill. Let P be the top of the

P

hill, and AB the base line. The vertical line through P will meet AB produced in C. AB 100 ft., CAP=30°, CBP=45°; the height CP is required. Let BC= x, and CP=y.

=

From (2), x = y. Substitution in (1) gives

y = (y +100) × .57735.

3. A flagstaff 30 ft. high stands on the top of a cliff, and from a point on a level with the base of the cliff the angles of elevation of the top and bottom of the flagstaff are observed to be 40° 20' and 38° 20', respectively. Find the height of the cliff.

Let BP be the flagstaff on the top of the cliff BL, and let C be the place of observation. BP 30 ft., LCB 38° 20', LCP = 40° 20'. Let CL=x,

4. At a point 180 ft. from a tower, and on a level with its base, the elevation of the top of the tower is found to be 65° 40.5'. What is the height of the tower?

5. From the top of a tower 120 ft. high the angle of depression of an object on a level with the base of the tower is 27° 43'. What is the distance of the object from the top and bottom of the tower?

6. From the foot of a post the elevation of the top of a column is 45°, and from the top of the post, which is 27 ft. high, the elevation is 30°. Find the height and distance of the column.

7. From the top of a cliff 120 ft. high the angles of depression of two boats, which are due south of the observer, are 20° 20′ and 68° 40'. Find the distance between the boats.

8. From the top of a hill 450 ft. high, the angle of depression of the top of a tower, which is known to be 200 ft. high, is 63° 20'. What is the distance from the foot of the tower to the top of the hill?

9. From the top of a hill the angles of depression of two consecutive mile-stones, which are in a direction due east, are 21° 30′ and 47° 40'. high is the hill ?

How

10. For an observer standing on the bank of a river, the angular elevation of the top of a tree on the opposite bank is 60°; when he retires 100 ft. from the edge of the river the angle of elevation is 30°. Find the height of the tree and the breadth of the river.

11. Find the distance in space travelled in an hour, in consequence of the earth's rotation, by an object in latitude 44° 20′. [Take earth's diameter equal to 8000 mi.]

12. At a point straight in front of one corner of a house, its height subtends an angle 34° 45', and its length subtends an angle 72° 30'; the height of the house is 48 ft. Find its length.

30. Problems requiring a knowledge of the points of the Mariner's Compass. The circle in the Mariner's Compass is divided into 32 equal parts, each part being thus equal to 360° ÷ 32, i.e. 114°. The points of division are named as indicated on the figure.

It will be observed that the points are named with reference to the points North, South, East, and West, which are called the cardinal points. Direction is indicated in a variety of ways. For instance, suppose C were the centre of the circle; then the point P in the figure is said to bear E.N.E. from C, or, from C the bearing of P is E.N.E. Similarly, C bears W.S.W. from P, or, the bearing of C from P is W.S.W. The point E.N.E. is 2 points North of East, and 6 points East of North. Accordingly, the phrases E. 221° N., N. 67° E., are sometimes used instead of E.N.E.

31. Mensuration. Let ABC be any triangle, and let the lengths of the sides opposite the angle A, B, C be denoted by a, b, c, respectively. From any vertex C draw CD at right angles to the opposite

side AB. It has been shown in arithmetic and geometry, that the area of a triangle is equal to one-half the product of the lengths of any side and the perpendicular drawn to it from the opposite vertex. [In (1) A is acute, in (2) A is obtuse.]

area ABC (Fig. 2) = † AB · DC;

=AB AC sin CAD;

=

be sin (1804).

It will be seen in Art. 45, that sin (180 — A) = sin A. Hence, the area of a triangle is equal to one-half the product of any two sides and the sine of their contained angle.

EXAMPLES.

1. Find the area of the triangle in which two sides are 31 ft. and 23 ft. and their contained angle 67° 30'.

2. Find area of triangle having sides 125 ft., 80 ft., contained angle 28° 35'.

3. Find area of triangle having sides 125 ft., 80 ft., contained angle 151° 25'. [Draw figures carefully for Exs. 2, 3.]

4. Find area of parallelogram two of whose adjacent sides are 243, 315 yd., and their included angle 35° 40'.