9. The angle of elevation of the top of a building from the foot of the tower is and the angle of elevation of the top of the tower from the foot of the building is . If the tower is 50 m high, find the height of the building.

Understand the Question Let represent the vertical tower of height and represent the vertical building of height . The horizontal ground distance between the building and the tower is . From the foot of the building (point ), the angle of elevation to the top of the tower () is . From the foot of the tower (point ), the angle of elevation to the top of the building () is . We use in to find the shared base distance , and then in to determine the building's height . Step 1 · Find the Distance Between the Building and Tower Let be the tower and be the building on the horizontal ground . In right Step 2 · Find the Height of the Building In right Answer > (or ) Common Mistakes Swapping the Angles of Elevation: Assigning to the building and to the tower. The taller structure (tower) must subtend the larger angle of elevation () over the same horizontal distance. Fraction Multiplication Error: Incorrectly computing as instead of .

Some Applications of Trigonometry | Exercise 9.1

Question 9

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.

Solution

Understand the Question

Let $\text{CD}$ represent the vertical tower of height $50\text{ m}$ and $\text{AB}$ represent the vertical building of height $h$ .
The horizontal ground distance between the building and the tower is $\text{BD}$ .
From the foot of the building (point $\text{B}$ ), the angle of elevation to the top of the tower ( $\text{C}$ ) is $60^\circ$ .
From the foot of the tower (point $\text{D}$ ), the angle of elevation to the top of the building ( $\text{A}$ ) is $30^\circ$ .
We use $\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$ in $\Delta \text{CBD}$ to find the shared base distance $\text{BD}$ , and then in $\Delta \text{ABD}$ to determine the building's height $\text{AB}$ .

Step 1 · Find the Distance Between the Building and Tower

Let $\text{CD} = 50\text{ m}$ be the tower and $\text{AB}$ be the building on the horizontal ground $\text{BD}$ . Diagram 1

In right $\Delta \text{CBD}$

\begin{aligned} \tan 60^\circ &= \dfrac{\text{CD}}{\text{BD}} \\[0.6em] \sqrt{3} &= \dfrac{50}{\text{BD}} \\[0.6em] \text{BD} &= \dfrac{50}{\sqrt{3}} \end{aligned}

Step 2 · Find the Height of the Building

In right $\Delta \text{ABD}$

\begin{aligned} \tan 30^\circ &= \dfrac{\text{AB}}{\text{BD}} \\[0.6em] \dfrac{1}{\sqrt{3}} &= \dfrac{\text{AB}}{\dfrac{50}{\sqrt{3}}} \\[1.1em] \text{AB} &= \dfrac{1}{\sqrt{3}} \times \dfrac{50}{\sqrt{3}} \\[0.6em] \text{AB} &= \dfrac{50}{3} = 16\dfrac{2}{3} \approx 16.67\text{ m} \end{aligned}

Answer

$\dfrac{50}{3}\text{ m}$ (or $16\dfrac{2}{3}\text{ m} \approx 16.67\text{ m}$ )

Common Mistakes

Swapping the Angles of Elevation: Assigning $60^\circ$ to the building and $30^\circ$ to the tower. The taller structure (tower) must subtend the larger angle of elevation ( $60^\circ$ ) over the same horizontal distance.
Fraction Multiplication Error: Incorrectly computing $\dfrac{50}{\sqrt{3}} \times \dfrac{1}{\sqrt{3}}$ as $50$ instead of $\dfrac{50}{3}$ .

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.