Home
Class 12
MATHS
From a point on a hillside of constant i...

From a point on a hillside of constant inclination , the angle of elevation of the top a flagstaff on its summit is observed to be `alpha` and a meters nears the top of the hill, it is `beta` .If h is the height of the flagstaff ,find the inclination of the hill to the horizon .

Text Solution

Verified by Experts

Let OP be the flagstaff of height h and `theta` inclination of the hill.

`Then angle ORP = alpha,angle OQP=beta and RQ =a.`
Also,`angle RPQ =beta-alpha .`
Now ,in triangle PQR ,using sine rule ,we get
`pr/(sin(180^@-beta))=(RQ)/sin(beta-alpha)`
`rArr PR = (asinbeta) /sin(beta-alpha) ` ...(1)
Also, `anglePOR= pi/2+ theta `
Now , in triangle PQR ,using sine rule , we get
`(PO)/sinalpha=(PR)/sin(pi/2+theta)`
`rArr PR =(hcostheta)/(sinalpha)=(asinbeta)/sin(beta-alpha)` (using(1))
`rArr theta = cos^-1[(asinalpha sin beta )/(hsin(beta-alpha))]`
Promotional Banner

Topper's Solved these Questions

  • HIGHT AND DISTANCE

    CENGAGE PUBLICATION|Exercise Exercises|18 Videos

  • HIGHT AND DISTANCE

    CENGAGE PUBLICATION|Exercise Archives|3 Videos

  • GRAPHS OF TRIGONOMETRIC FUNCTIONS

    CENGAGE PUBLICATION|Exercise Exercises|22 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise COMPREHENSION TYPE|2 Videos

Similar Questions

Explore conceptually related problems

The angle of elevation of the top of a tower a point A due south of it is 30^@ and from a point B due west of it is 45^@ .If the height of the tower is 100 meters ,then find the distance AB.

The angle of elevation of the top of a tower from a point O on the ground, which is 450 m away from the foot of the tower, is 40^(@) . Find the height of the tower.

From the bottom of a pole of height h, the angle of elevation of the top of a tower is alpha . The pole subtends an angle beta at the top of the tower. find the height of the tower.

From the top and bottom of a cliff the angles of depression and elevation of the top of a pillar 60ft high are observed to be 30^@ and 60^@ respectively. Find the height of the cliff.

If the angle of elevation of the top of a tower from two points distant a and b from the base and in the same straight line with it are comple mentary, then the height of the tower is____

From the bottom of a pole of height h the angle of elevation of the top of a tower is alpha and the pole subtends an angle beta at the top of the tower . Prove that the height of the tower is (h cot(alpha - beta))/(cot(alpha - beta) - cot alpha) .

The angle of elevation of the top of the tower observed from each of the three points A,B,C on the ground, forming a triangle is the same angle prop . If R is the circum-radius of the triangle ABC, then the height of the tower is

The height of the mobile tower is 40sqrt3 metre. The angle of elevation of the top of the tower from a point a distance 120 metre from the foot of the tower is

From a point on the horizontal line through the foot of a chimney the angle of elevation of the top of the chimney is 30^@ . and the angle of elevation is 60^@ at a point on the same straight line 50 metres nearer to the chimney. What is the height of the chimney?

From a point on the roof of five-storied building the angle of elevation of the top of a monument and that of angle of depression of the foot of the monument are 60^(@) and 30^(@) respectively. If the height of the building is 16 metres, calculate the height of the monument and the distance of the building from the monument.