Some Applications of Trigonometry Set-1

Q1. From a point on level ground 30 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is $30^\circ$. What is the height of the tower?

Q2. Two vertical poles of heights $45\text{ m}$ and $27\text{ m}$ stand on the same straight line on level ground. From a point between their bases the angles of elevation to their tops are $60^\circ$ and $30^\circ$ respectively. What is the distance between the two poles?

Q3. From the top of a lighthouse $100\text{ m}$ high, the angles of depression of two ships on opposite sides are $30^\circ$ and $45^\circ$. The ships and the lighthouse are on the same horizontal plane. The distance between the two ships is

Q4. Two poles of equal height stand on opposite sides of a straight road which is $50\text{ m}$ wide. From a point on the road between their bases, the angles of elevation of their tops are $60^\circ$ and $30^\circ$ respectively. Find the distance of the point from the nearer pole and the common height of the poles.

Q5. From two points on the same straight line towards a vertical tower, the angles of elevation of its top are $45^\circ$ (closer point) and $30^\circ$ (farther point). The two observation points are $30\text{ m}$ apart. The height of the tower is

Q6. A ladder 10 m long leans against a vertical wall making an angle of $60^\circ$ with the horizontal. How high up the wall does the ladder reach?

Q7. From two points on the same straight line towards a vertical tower, the angles of elevation of its top are $60^\circ$ (from the nearer point) and $30^\circ$ (from the farther point). If the distance between the two points is $10\ \text{m}$, find the height of the tower.

Q8. From two points $A$ and $B$ on the same side of a vertical tower, the angles of elevation of the top are $30^\circ$ (from $A$) and $45^\circ$ (from $B$). If $B$ is nearer to the tower and $AB=20\ \text{m}$, the height of the tower is

Q9. A man walks towards a vertical tower of height $10\sqrt{3}\ \text{m}$. While walking, the angle of elevation of the top of the tower increases from $30^\circ$ to $60^\circ$ in $10\ \text{s}$. What is the speed of the man (assume straight line motion towards the tower)?

Q10. A tree breaks at some height and the top touches the ground. The broken part makes an angle of $30^\circ$ with the horizontal and the point where the top touches the ground is $15\ \text{m}$ from the base of the tree. Find the original height of the tree.

Q11. A vertical tower stands on level ground. An observer at a point $20\,$m from the base measures the angle of elevation of the top of the tower as $30^\circ$. What is the height of the tower?

Q12. Two poles of heights $10\,$m and $15\,$m stand vertically on level ground and their bases are $20\,$m apart. A point on the line joining their bases sees the tops of both poles under the same angle of elevation. How far is this point from the foot of the $10\,$m pole?

Q13. A lighthouse of unknown height stands on a straight shore. From a point $A$ on the shore the angle of elevation of its top is $45^\circ$. Moving $100\,$m directly away from the lighthouse to point $B$, the angle of elevation becomes $30^\circ$. What is the height of the lighthouse?

Q14. The top of a tower is observed from ground level at an angle of elevation $30^\circ$. A man of height $1.6\,$m then walks $12\,$m towards the tower and at the new position the angle of elevation to the top is $60^\circ$. Find the height of the tower (in metres).

Q15. From the top of a vertical tower, the angles of depression to two points $A$ and $B$ on the horizontal ground are $60^\circ$ and $30^\circ$ respectively. If the distance between $A$ and $B$ is $10\,$m and both points lie on the same straight line passing under the tower, what is the height of the tower?