Amit M Agarwal Solutions for Chapter: Properties of Triangles, Heights and Distances, Exercise 3: Work Book Exercise 10.3

A piece of paper in the shape of a sector of a circle of radius $10 \mathrm{cm}$ and of angle $216°$ just covers the lateral surface of a right circular cone of vertical angle $2\theta$. The $\sin \theta$ is

The shadow of a tower standing on a level ground is $x m$ long when the Sun's altitude is $30°,$ while it is $y m$ long when the altitude is $60°.$ If the height of the tower is $45\times \frac{\sqrt{3}}{2} m,$ then $x-y$ is

A vertical lamp-post of height $9 \mathrm{m}$ stands at the corner of a rectangular field. The angle of elevation of its top from the farthest corner is $30°$, while from another corner it is $45°$. The area of the field is

A vertical lamp-post, $6 \mathrm{m}$ high, stands at a distance of $2 \mathrm{m}$ from a wall, $4 \mathrm{m}$ high. A $1.5 \mathrm{m}$ tall man starts to walk away from the wall on the other side of the wall, in line with the lamp-post. The maximum distance to which the man can walk remaining in the shadow is

A man standing between two vertical posts finds that the angle subtended at his eyes by the tops of the posts is a right angle. If the heights of the two posts are two times and four times the height of the man and the distance between them is equal to the length of the longer post, then the ratio of the distances of the man from the shorter and the longer post is

A $6$ ft tall man finds that the angle of elevation of the top of a $24$ ft high pillar and the angle of depression of its foot are complementary angles. The distance of the man from the pillar is

A rocket of height $h \mathrm{m}$ is fired vertically upwards. Its velocity at time $t$ seconds is $(2t+3)\mathrm{m}/\mathrm{s}$. If the angles of elevation of the top of the rocket from a point on the ground after $1 s$ of firing is $\frac{\pi }{6}$ and after $3 s$ it is $\frac{\pi }{3}$, then the distance of the point from the rocket is

The angle of elevation of the top of a vertical pole, when observed from each vertex of a regular hexagon is $\frac{\pi }{3}$. If area of the circle circumscribing the hexagon is $A$ $\mathrm{sq} \mathrm{m}$, then area of the hexagon is