Measurements in more than one plane.

201. In Art. 199 the base line AB was measured directly towards the object. If this is not possible we may proceed as follows.

From A measure a base line AB in any convenient direction in the horizontal plane. At A observe the two angles PAB and PAC; and at B observe the angle PBA.

Let LPAB=a, LPAC=ß,
LPBA=y,
AB=a, PC=x.

From A PAC,

X= = PA sin B.

a is

202. To shew how to find the distance between two inaccessible objects.

Let P and Q be the objects.

Take any two convenient stations A and B in the same horizontal plane, and measure the distance between them.

At A observe the angles PAQ and QAB. Also if AP, AQ, AB are not in the same plane, measure the angle PAB.

At B observe the angles ABP A and ABQ.

In ▲PAB, we know LPAB, LPBA, and AB;

so that AP may be found.

In AQAB, we know ▲QAB, ▲ QBA, and AB;
so that AQ may be found.

In APAQ, we know AP, AQ, and ▲ PAQ;

so that PQ may be found.

H. K. E. T.

Example 1. The angular elevation of a tower CD at a place A due South of it is 30°, and at a place B due West of A the elevation

is 18°. If AB=a, shew that the height of the tower is

a

√2+2√5

From the right-angled triangle DCA, AC=x cot 30°.
From the right-angled triangle DCB, BC=x cot 18°.
But BAC is a right angle,

N

x2 {(√5+1)2-4}=a2;
.. x2 (2+2/5)=a2,

which gives the height required.

Example 2. A hill of inclination 1 in 5 faces South. Shew that a road on it which takes a N.E. direction has an inclination 1 in 7.

Let AD running East and West be the ridge of the hill, and let ABFD be a vertical plane through AD. Let C be a point at the foot of the hill, and ABC a section made by a vertical plane running North and South. Draw CG in a N.E. direction in the horizontal plane and let it meet BF in G; draw GH parallel to BA; then if CH is joined it will represent the direction of the road.

Since the inclination of CA is 1 in 5, we may take AB=a, and AC=5a, so that BC2=24a2.

Since CBG is a right-angled isosceles triangle,

CG2=2CB2=48a2.

Hence in the right-angled triangle CGH,

CH2=48a2+ a2=49a2;

.. CH=7a=7GH.

Thus the slope of the road is 1 in 7.

EXAMPLES. XVII. c.

1. The elevation of a hill at a place P due East of it is 45°, and at a place due South of P the elevation is 30°. If the distance from P to Q is 500 yards, find the height of the hill in feet.

2. The elevation of a spire at a point A due West of it is 60°, and at point B due South of A the elevation is 30°. If the spire is 250 feet high, find the distance between A and B.

3. A river flows due North, and a tower stands on its left bank. From a point A up-stream and on the same bank as the tower the elevation of the tower is 60°, and from a point B just opposite on the other bank the elevation is 45°. If the tower is 360 feet high, find the breadth of the river.

4. The elevation of a steeple at a place A due S. of it is 45°, and at a place B due W. of A the elevation is 15°. If AB=2a, shew that the height of the steeple is a (34 — 374).

5. A person due S. of a lighthouse observes that his shadow cast by the light at the top is 24 feet long. On walking 100 yards due E. he finds his shadow to be 30 feet long. Supposing him to be 6 feet high, find the height of the light from the ground.

6. The angles of elevation of a balloon from two stations a mile apart and from a point halfway between them are observed to be 60°, 30°, and 45° respectively. Prove that the height of the balloon is 440/6 yards.

[If AD is a median of the triangle ABC,

then 2AD2+2BD2=AB2+AC2.]

7. At each end of a base of length 2a, the angular elevation of a mountain is 0, and at the middle point of the base the elevation is . Prove that the height of the mountain is

a sin o sin cosec (+0) cosec (p − 0).

8. Two vertical poles, whose heights are a and b, subtend the same angle a at a point in the line joining their feet. If they subtend angles B and y at any point in the horizontal plane at which the line joining their feet subtends a right angle, prove

that

(a+b)2 cot2 a=a2 cot2 ẞ+b2 cot2 y.

9. From the top of a hill a person finds that the angles of depression of three consecutive milestones on a straight level road are a, ß, y. Shew that the height of the hill is

5280/2/cot2 a-2 cot2 B+cot2 y feet.

10. Two chimneys AB and CD are of equal height. A person standing between them in the line AC joining their bases observes the elevation of the one nearer to him to be 60°. After walking 80 feet in a direction at right angles to AC he observes their elevations to be 45° and 30°: find their height and distance apart.

11. Two persons who are 500 yards apart observe the bearing and angular elevation of a balloon at the same instant. One finds the elevation 60° and the bearing S. W., the other finds the elevation 45° and the bearing W. Find the height of the balloon.

12. The side of a hill faces due S. and is inclined to the horizon at an angle a. A straight railway upon it is inclined at an angle ẞ to the horizon: if the bearing of the railway be r degrees E. of N., shew that cos a = cot a tan ß.

EXAMPLES. XVII. d.

[In the following examples the logarithms are to be taken from the Tables.]

1. A man in a balloon observes that two churches which he knows to be one mile apart subtend an angle of 11° 25′ 20′′ when he is exactly over the middle point between them: find the height of the balloon in miles.

2. There are three points A, B, C in a straight line on a level piece of ground. A vertical pole erected at C has an elevation of 5° 30' from A and 10° 45′ from B. If AB is 100 yards, find the height of the pole and the distance BC.

3. The angular altitude of a lighthouse seen from a point on the shore is 12° 31′ 46′′, and from a point 500 feet nearer the altitude is 26° 33′ 55′′: find its height above the sea-level.

4. From a boat the angles of elevation of the highest and lowest points of a flagstaff 30 ft. high on the edge of a cliff are 46° 12′ and 44° 13': find the height and distance of the cliff.

5. From the top of a hill the angles of depression of two successive milestones on level ground, and in the same vertical plane as the observer, are 5° and 10°. Find the height of the hill in feet and the distance of the nearer milestone in miles.

6. An observer whose eye is 15 feet above the roadway finds that the angle of elevation of the top of a telegraph post is 17° 18′ 35′′, and that the angle of depression of the foot of the post is 8° 32′ 15": find the height of the post and its distance from the observer.

7. Two straight railroads are inclined at an angle of 20° 16'. At the same instant two engines start from the point of intersection, one along each line; one travels at the rate of 20 miles an hour at what rate must the other travel so that after 3 hours the distance between them shall be 30 miles?

8. An observer finds that from the doorstep of his house the elevation of the top of a spire is 5a, and that from the roof above the doorstep it is 4a. If h be the height of the roof above the doorstep, prove that the height of the spire above the doorstep and the horizontal distance of the spire from the house are respectively

h cosec a cos 4a sin 5a and h cosec a cos 4a cos 5a. If h=39 feet, and a=7° 17′ 39", calculate the height and distance.