Some Applications of Trigonometry | Exercise 9.1

Question 7

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.

Solution

Understand the Question

Let the building be $\text{BC} = 20\text{ m}$ and the transmission tower on top of it be $\text{CD}$ .
From a point $\text{A}$ on the ground, the angle of elevation of the bottom of the tower (point $\text{C}$ ) is $45^\circ$ and the top of the tower (point $\text{D}$ ) is $60^\circ$ .
In right-angled triangle $\Delta \text{ABC}$ , use $\tan 45^\circ$ to find the horizontal distance $\text{AB}$ .
In right-angled triangle $\Delta \text{ABD}$ , use $\tan 60^\circ$ to find the total height $\text{BD}$ .
The height of the tower is given by $\text{CD} = \text{BD} - \text{BC}$ .

Step 1 · Find the Distance from Observation Point to Building

Let $\text{BC} = 20\text{ m}$ be the height of the building, $\text{CD}$ be the height of the tower, and $\text{A}$ be the observation point on the ground. Diagram 1

In right $\Delta \text{ABC}$

\begin{aligned} \tan 45^\circ &= \dfrac{\text{BC}}{\text{AB}} \\[0.6em] 1 &= \dfrac{20}{\text{AB}} \\[0.6em] \text{AB} &= 20\text{ m} \end{aligned}

Step 2 · Find the Total Height

In right $\Delta \text{ABD}$ , where $\text{BD}$ is the total height from the ground to the top of the tower

\begin{aligned} \tan 60^\circ &= \dfrac{\text{BD}}{\text{AB}} \\[0.6em] \sqrt{3} &= \dfrac{\text{BD}}{20} \\[0.6em] \text{BD} &= 20\sqrt{3}\text{ m} \\[0.6em] &= 20 \times 1.732 \\[0.6em] &= 34.64\text{ m} \end{aligned}

Step 3 · Calculate the Height of the Tower

The height of the tower is the difference between total height $\text{BD}$ and building height $\text{BC}$

\begin{aligned} \text{Height of tower } (\text{CD}) &= \text{BD} - \text{BC} \\[0.6em] &= 20\sqrt{3} - 20 \\[0.6em] &= 20(\sqrt{3} - 1)\text{ m} \\[0.6em] &= 20(1.732 - 1) \\[0.6em] &= 20(0.732) \\[0.6em] &= 14.64\text{ m} \end{aligned}

Answer

$20(\sqrt{3} - 1)\text{ m} \approx 14.64\text{ m}$

Common Mistakes

Angle Swapping: Assigning $60^\circ$ to the bottom of the tower and $45^\circ$ to the top. The top of the tower is higher, so its angle of elevation must be larger ( $60^\circ$ ).
Base Length Assumption: Assuming the height of the tower directly from $\tan 60^\circ$ without subtracting the height of the building ( $20\text{ m}$ ).
Approximation Errors: Forgetting units ( $\text{m}$ ) or substituting $\sqrt{3} \approx 1.732$ incorrectly.

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.

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.