10. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are and , respectively. Find the height of the poles and the distances of the point from the poles.

Understand the Question Two vertical poles of equal height stand on opposite sides of a road of width . From a point on the road between them, the angles of elevation to the tops of the poles are and . If the distance of point from one pole is , then its distance from the other pole is . Using the trigonometric ratio in both right triangles gives two expressions for , which can be equated to find and . Step 1 · Define variables and set up equations Let and be the two poles of equal height , and let be the observation point on the road . Let , so . In right In right Step 2 · Calculate the distances of the point from the poles Equating expressions for from equations and Distance from the first pole . Distance from the second pole Step 3 · Calculate the height of the poles Substitute into equation Answer > Height of the poles ; distances of the point from the poles are and . Common Mistakes Angle-Distance Mismatch: The closer point to a pole has the steeper angle of elevation (), so the pole with angle corresponds to the shorter distance , not . Segment Division Error: Setting the two distances as and instead of and . Trigonometric Values: Confusing with .

Some Applications of Trigonometry | Exercise 9.1

Question 10

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$ , respectively. Find the height of the poles and the distances of the point from the poles.

Solution

Understand the Question

Two vertical poles of equal height $h$ stand on opposite sides of a road of width $80\text{ m}$ .
From a point $P$ on the road between them, the angles of elevation to the tops of the poles are $60^\circ$ and $30^\circ$ .
If the distance of point $P$ from one pole is $x$ , then its distance from the other pole is $80 - x$ .
Using the trigonometric ratio $\tan \theta = \dfrac{\text{Opposite}}{\text{Adjacent}} = \dfrac{\text{Height}}{\text{Distance}}$ in both right triangles gives two expressions for $h$ , which can be equated to find $x$ and $h$ .

Step 1 · Define variables and set up equations

Let $\text{AB}$ and $\text{CD}$ be the two poles of equal height $h$ , and let $P$ be the observation point on the road $\text{BD} = 80\text{ m}$ . Let $\text{BP} = x$ , so $\text{PD} = 80 - x$ . Diagram 1

In right $\triangle \text{ABP}$

\begin{aligned} \tan 60^\circ &= \dfrac{\text{AB}}{\text{BP}} \\[0.6em] \sqrt{3} &= \dfrac{h}{x} \\[0.6em] h &= x\sqrt{3} \quad \dots (1) \end{aligned}

In right $\triangle \text{CDP}$

\begin{aligned} \tan 30^\circ &= \dfrac{\text{CD}}{\text{PD}} \\[0.6em] \dfrac{1}{\sqrt{3}} &= \dfrac{h}{80 - x} \\[0.6em] h &= \dfrac{80 - x}{\sqrt{3}} \quad \dots (2) \end{aligned}

Step 2 · Calculate the distances of the point from the poles

Equating expressions for $h$ from equations $(1)$ and $(2)$

\begin{aligned} x\sqrt{3} &= \dfrac{80 - x}{\sqrt{3}} \\[0.6em] x\sqrt{3} \times \sqrt{3} &= 80 - x \\[0.6em] 3x &= 80 - x \\[0.6em] 3x + x &= 80 \\[0.6em] 4x &= 80 \\[0.6em] x &= \dfrac{80}{4} = 20\text{ m} \end{aligned}

Distance from the first pole $\text{BP} = 20\text{ m}$ .

Distance from the second pole $\text{PD}$

80 - x = 80 - 20 = 60\text{ m}

Step 3 · Calculate the height of the poles

Substitute $x = 20$ into equation $(1)$

\begin{aligned} h &= x\sqrt{3} \\[0.6em] &= 20\sqrt{3}\text{ m} \end{aligned}

Answer

Height of the poles $= 20\sqrt{3}\text{ m}$ ; distances of the point from the poles are $20\text{ m}$ and $60\text{ m}$ .

Common Mistakes

Angle-Distance Mismatch: The closer point to a pole has the steeper angle of elevation ( $60^\circ$ ), so the pole with angle $60^\circ$ corresponds to the shorter distance $20\text{ m}$ , not $60\text{ m}$ .
Segment Division Error: Setting the two distances as $x$ and $80 + x$ instead of $x$ and $80 - x$ .
Trigonometric Values: Confusing $\tan 60^\circ = \sqrt{3}$ with $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$ .

More questions in Exercise 9.1

A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is $30^\circ$ (see Fig. 9.11).

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of $30^\circ$ to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m, and inclined at an angle of $60^\circ$ to the ground. What should be the length of the slide in each case?

The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is $30^\circ$ . Find the height of the tower.

A kite is flying at a height of $60\text{ m}$ above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is $60^\circ$ . Find the length of the string, assuming that there is no slack in the string.

A $1.5\text{ m}$ tall boy is standing at some distance from a $30\text{ m}$ tall building. The angle of elevation from his eyes to the top of the building increases from $30^\circ$ to $60^\circ$ as he walks towards the building. Find the distance he walked towards the building.

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are $45^\circ$ and $60^\circ$ respectively. Find the height of the tower.

A statue, $1.6\text{ m}$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^\circ$ and from the same point the angle of elevation of the top of the pedestal is $45^\circ$ . Find the height of the pedestal.

The angle of elevation of the top of a building from the foot of the tower is $30^\circ$ and the angle of elevation of the top of the tower from the foot of the building is $60^\circ$ . If the tower is 50 m high, find the height of the building.

Q10

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$ , respectively. Find the height of the poles and the distances of the point from the poles.

Q11

A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60^\circ$ . From another point 20 m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is $30^\circ$ (see Fig. 9.12). Find the height of the tower and the width of the canal.

Q12

From the top of a $7\text{ m}$ high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$ . Determine the height of the tower.

Q13

As observed from the top of a $75\text{ m}$ high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $45^\circ$ . If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Q14

A $1.2\text{ m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\text{ m}$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$ . After some time, the angle of elevation reduces to $30^\circ$ (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Q15

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$ , which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be $60^\circ$ . Find the time taken by the car to reach the foot of the tower from this point.