A car is travelling on a straight road leading to a tower. From a point at a distance of 500 m from the tower, as seen by the driver, the angle of elevation of the top of the tower is `30^@`. After driving towards the tower for 10 seconds, the angle of elevation of the top of the tower as seen by the driver is found to be `60^@`. Then the speed of the car is

Options

A. 135 km/hr
B. 110 km/hr
C. 120 km/hr
D. 90 km/hr

Answer

To solve the problem step by step, we will analyze the situation using trigonometry.

### Step 1: Understand the scenario
We have a tower and a car moving towards it. The initial distance from the car to the tower is 500 m, and at this point, the angle of elevation to the top of the tower is 30 degrees.

### Step 2: Set up the first triangle
Let:
- \( AB \) = height of the tower
- \( BC \) = initial distance from the car to the tower = 500 m
- \( \angle ABC = 30^\circ \)

Using the tangent function for the first position:
\[ \tan(30^\circ) = \frac{AB}{BC} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{500} \]
From this, we can solve for \( AB \):
\[ AB = \frac{500}{\sqrt{3}} \quad \text{(Equation 1)} \]

### Step 3: Set up the second triangle
After driving for 10 seconds, the angle of elevation changes to 60 degrees. Let \( CK \) be the distance travelled by the car in 10 seconds. The new distance from the car to the tower is \( BC - CK \).

Using the tangent function for the second position:
\[ \tan(60^\circ) = \frac{AB}{(500 - CK)} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we can write:
\[ \sqrt{3} = \frac{AB}{500 - CK} \]
Substituting \( AB \) from Equation 1:
\[ \sqrt{3} = \frac{\frac{500}{\sqrt{3}}}{500 - CK} \]
Cross-multiplying gives:
\[ \sqrt{3}(500 - CK) = \frac{500}{\sqrt{3}} \]
Multiplying both sides by \( \sqrt{3} \):
\[ 3(500 - CK) = 500 \]
Expanding and rearranging:
\[ 1500 - 3CK = 500 \]
\[ 3CK = 1000 \]
\[ CK = \frac{1000}{3} \quad \text{(Equation 2)} \]

### Step 4: Calculate the speed of the car
The car travels \( CK \) in 10 seconds. Speed is given by:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{CK}{10} \]
Substituting \( CK \) from Equation 2:
\[ \text{Speed} = \frac{\frac{1000}{3}}{10} = \frac{1000}{30} = \frac{100}{3} \text{ m/s} \]

### Step 5: Convert speed from m/s to km/h
To convert from meters per second to kilometers per hour, we multiply by \( \frac{18}{5} \):
\[ \text{Speed in km/h} = \frac{100}{3} \times \frac{18}{5} = \frac{1800}{15} = 120 \text{ km/h} \]

### Final Answer
The speed of the car is **120 km/h**.

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