Some Applications of Trigonometry — Class 10 Mathematics (Standard) PYQs

A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as $60^{\circ }$. If $AB=5\sqrt{3}\text{m}$, then length of the wire is

Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is $54$ metres away from the water tank. From a window (W) of the building, the angle of elevation of top of the tank is $45^{\circ }$ and angle of depression of its foot is $30^{\circ }$.
(i) Write a relation between $d$ (the height of window) and $y$.
(ii) Determine the value of $h$.
(iii) (a) Determine height of the water tank.
OR
(iii) (b) Find the value of $x$ and height of the window above ground level.

A kite is flying at a height of $150\text{m}$ from the ground. It is attached to a string inclined at an angle of $30^{\circ }$ to the horizontal. The length of the string is :

Case Study - 2
Amrita stood near the base of a lighthouse, gazing up at its towering height. She measured the angle of elevation to the top and found it to be $60^{\circ }$. Then, she climbed a nearby observation deck, $40$ metres higher than her original position and noticed the angle of elevation to the top of lighthouse to be $45^{\circ }$.
Based on the above given information, answer the following questions :
(i) If $CD$ is $h$ metres, find the distance $BD$ in terms of $h$.
(ii) Find distance $BC$ in terms of $h$.
(iii) (a) Find the height $CE$ of the lighthouse [Use $\sqrt{3}=1.73$]
OR
(b) Find distance $AE$, if $AC=100\text{m}$.

A ladder $14\text{m}$ long leans against a wall. If the foot of the ladder is $7\text{m}$ from the wall, then the angle of elevation of the top of the wall is :

(a) Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are $60^{\circ }$ and $45^{\circ }$, respectively. If the distance between the ships is $100\left(\frac{1+\sqrt{3}}{\sqrt{3}}\right)\text{m}$, then find the height of the lighthouse.
OR
(b) The angles of depression of the top and the bottom of an $8\text{m}$ tall building from the top of another multistoried building are $30^{\circ }$ and $45^{\circ }$, respectively. Find the height of the multistoried building and the distance between the two buildings.

(a) The angle of elevation of an airborne helicopter from a point A on the ground is $45^{\circ }$. After a flight of $15$ seconds, the angle of elevation of the helicopter changes to $30^{\circ }$. If the helicopter is flying at a constant height of $2000\text{m}$, find the speed of the helicopter. (Take $\sqrt{3}=1.732$)
OR
(b) A girl $1.5\text{m}$ tall is standing at some distance from a $30\text{m}$ high tower. The angle of elevation from her eye to the top of the tower increases from $30^{\circ }$ to $60^{\circ }$ as she walks towards the tower. Find the distance she walked towards the tower.

A pole $6\text{m}$ high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point P on the ground is $60^{\circ }$ and the angle of depression of the point P from the top of the tower is $45^{\circ }$. Find the height of the tower and the distance of point P from the foot of the tower. (Use $\sqrt{3}=1.73$)

From the top of a building $60\text{m}$ high, the angles of depression of the top and bottom of the vertical lamp post are observed to be $30^{\circ }$ and $60^{\circ }$ respectively.
(i) Find the horizontal distance between the building and the lamp post.
(ii) Find the distance between the tops of the building and the lamp post.

From the top of a $15\text{m}$ high building, the angle of elevation of the top of a tower is found to be $30^{\circ }$. From the bottom of the same building, the angle of elevation of the top of the tower is found to be $60^{\circ }$. Find the height of the tower and the distance between tower and the building.