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Q8:
A statue, $1.6$ 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 $60°$ and from the same point the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.
Solution :
Given:
- Height of the statue ($AB$) = $1.6$ m.
- The statue stands on a pedestal ($BC$).
- Angle of elevation from a point $D$ on the ground to the top of the statue ($A$) is $\angle ADB = 60^\circ$.
- Angle of elevation from the same point $D$ to the top of the pedestal ($B$) is $\angle CDB = 45^\circ$.
To Find:
The height of the pedestal ($BC = h$ meters).
Step 1: Define Variables and Assumptions
Let $BC = h$ be the height of the pedestal in meters.
Let $CD = x$ be the distance from the point $D$ to the base of the pedestal $C$ in meters.
The total height of the statue and pedestal is $AC = AB + BC = 1.6 + h$.
Step 2: Analyze the smaller triangle $\triangle BCD$
In the right-angled triangle $\triangle BCD$, the angle $\angle CDB = 45^\circ$.
Using the trigonometric ratio $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$:
$\tan(45^\circ) = \frac{BC}{CD}$
Since $\tan(45^\circ) = 1$:
$1 = \frac{h}{x} \implies x = h$ --- (Equation 1)
Step 3: Analyze the larger triangle $\triangle ACD$
In the right-angled triangle $\triangle ACD$, the angle $\angle ADB = 60^\circ$.
Using the trigonometric ratio $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$:
$\tan(60^\circ) = \frac{AC}{CD}$
Substitute $AC = 1.6 + h$ and $CD = x$:
$\sqrt{3} = \frac{1.6 + h}{x}$ --- (Equation 2)
Step 4: Solve for $h$
Substitute $x = h$ (from Equation 1) into Equation 2:
$\sqrt{3} = \frac{1.6 + h}{h}$
Multiply both sides by $h$:
$h\sqrt{3} = 1.6 + h$
Rearrange the terms to isolate $h$:
$h\sqrt{3} - h = 1.6$
$h(\sqrt{3} - 1) = 1.6$
$h = \frac{1.6}{\sqrt{3} - 1}$
Step 5: Rationalize the denominator
To simplify, multiply the numerator and denominator by the conjugate $(\sqrt{3} + 1)$:
$h = \frac{1.6(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}$
Using the identity $(a-b)(a+b) = a^2 - b^2$:
$h = \frac{1.6(\sqrt{3} + 1)}{3 - 1}$
$h = \frac{1.6(\sqrt{3} + 1)}{2}$
$h = 0.8(\sqrt{3} + 1)$
Using $\sqrt{3} \approx 1.732$:
$h = 0.8(1.732 + 1) = 0.8(2.732) = 2.1856$ m.
Final Answer: The height of the pedestal is $0.8(\sqrt{3} + 1)$ m or approximately $2.186$ m.
More Questions from Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1
- Q1: 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 $30°$ (see Fig. 9.11).
- 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 $60°$ and $30°$, 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 $60°$. 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 $30°$ (see Fig. 9.12). Find the height of the tower and the width of the canal.
- Q12: From the top of a $7$ m high building, the angle of elevation of the top of a cable tower is $60°$ and the angle of depression of its foot is $45°$. Determine the height of the tower.
- Q13: As observed from the top of a $75$ m high lighthouse from the sea-level, the angles of depression of two ships are $30°$ and $45°$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
- Q14: A $1.2$ m tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2$ m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60°$. After some time, the angle of elevation reduces to $30°$ (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 $30°$, 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 $60°$. Find the time taken by the car to reach the foot of the tower from this point.
- Q2: A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30°$ 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.
- Q3: 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 $30°$ 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 $60°$ to the ground. What should be the length of the slide in each case?
- Q4: 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 $30°$. Find the height of the tower.
- Q5: A kite is flying at a height of $60$ 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 $60°$. Find the length of the string, assuming that there is no slack in the string.
- Q6: A $1.5$ m tall boy is standing at some distance from a $30$ m tall building. The angle of elevation from his eyes to the top of the building increases from $30°$ to $60°$ as he walks towards the building. Find the distance he walked towards the building.
- Q7: 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 $45°$ and $60°$ respectively. Find the height of the tower.
- Q9: The angle of elevation of the top of a building from the foot of the tower is $30°$ and the angle of elevation of the top of the tower from the foot of the building is $60°$. If the tower is $50$ m high, find the height of the building.
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