A radar detect an aeroplane in the sky at an altitude of 5 km , at an angle of elevation of 45 .
Find the distance of the aeroplane from the radar .

Options

A. `5sqrt(2)`
B. 4 km
C. 5 km
D. 6 km

Answer

To find the distance of the aeroplane from the radar, we can use trigonometric principles. Let's break down the solution step by step.

### Step 1: Understand the problem
We have a radar detecting an aeroplane at an altitude of 5 km, and the angle of elevation from the radar to the aeroplane is 45 degrees. We need to find the distance from the radar to the aeroplane.

### Step 2: Draw a diagram
Draw a right triangle where:
- The vertical side (opposite side) represents the altitude of the aeroplane, which is 5 km.
- The horizontal side (adjacent side) represents the distance from the radar to the point directly below the aeroplane on the ground.
- The hypotenuse represents the distance from the radar to the aeroplane.

### Step 3: Identify the trigonometric function
Since we are given the angle of elevation (45 degrees) and the opposite side (altitude of the aeroplane), we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, \(\theta = 45^\circ\), opposite = 5 km, and we need to find the adjacent side.

### Step 4: Calculate the adjacent side
Using the tangent function:
\[ \tan(45^\circ) = 1 \]
This means:
\[ 1 = \frac{5 \text{ km}}{\text{adjacent}} \]
Thus, the adjacent side (distance on the ground) is also 5 km.

### Step 5: Use the Pythagorean theorem to find the hypotenuse
Now we can find the hypotenuse (the distance from the radar to the aeroplane) using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Substituting the values:
\[ \text{hypotenuse}^2 = (5 \text{ km})^2 + (5 \text{ km})^2 \]
\[ \text{hypotenuse}^2 = 25 + 25 = 50 \]
\[ \text{hypotenuse} = \sqrt{50} = 5\sqrt{2} \text{ km} \]

### Conclusion
The distance of the aeroplane from the radar is \(5\sqrt{2}\) km.

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