MEDIUM
JEE Main
IMPORTANT
Earn 100

A pyramid has for base a square of side a; its vertex lies on a line through the middle point of the base and perpendicular to it, and at a distance h from it; prove that the angle α between the two lateral faces is given by the equation sin⁡α=2h2a2+4h2a2+4h2.

Important Questions on Heights and Distances

HARD
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 5
A person on a ship A observes another ship B leaving a harbour, whose bearing is then N.W.. After 10 minutes A, having sailed one km. N.E., sees B due west and the harbour then bears  60°west of north. After another 10 minutes B is observed to bear S.W.. Find the distance between A and B at the first observation and also the direction and rate of B.
HARD
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 29
The extremity of the shadow of a flagstaff, which is 6m high and stands on the top of a pyramid on a square base, just reaches the side of the base and is distant 56m and 8m, respectively from the extremities of that side. Find the sun’s altitude if the height of the pyramid be 34 metres.
MEDIUM
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 30
The extremity of the shadow of a flagstaff, which is 6 metres high and stands on the top of a pyramid on a square base, just reaches the side of the base and is distant x feet and y feet respectively from the ends of that side. Prove that the height of the pyramid is x2+y22tanα-6, where α is the elevation of the sun.
HARD
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 31
The angle of elevation of a cloud from a point h metres above a lake is α, and the angle of depression of its reflection in the lake is β. Prove that its height is hsinβ+αsinβ-α.
MEDIUM
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 33
An isosceles triangle of wood is place in a vertical plane, vertex upwards, and faces the sun. If  2a be the base of the triangle, h its height and 30° the altitude of the sun, prove that the tangent of the angle at the apex of the shadow is 2ah33h2-a2.
HARD
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 34

A rectangular target faces due south, being vertical and standing on a horizontal plane. Compare the area of the target with that of its shadow on the ground when the sun is β° form the south at an altitude of α°.

MEDIUM
JEE Main
IMPORTANT
Plane Trigonometry Part 1>Chapter 13 - Heights and Distances>Examples XXXIV>Q 35
A spherical ball of diameter δ, subtends an angle α at a man’s eye when the elevation of its center is β, prove that the height of the center of the ball is 12δsinβ.cscα2.