A man standing in one corner of a square football field observes that the angle subtended by a pole in the corner just diagonally opposite to this corner is `60^@`. When he retires 80 m from the corner, along the same straight line, he finds the angle to be `30^@`. The length of the field is

Options

A. a)20 m
B. b)`40 sqrt 2m`
C. c)`40 m `
D. d)`20 sqrt 2m`

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Geometry
We have a square football field, and a man is standing at one corner (let's call it point A). The pole is at the diagonally opposite corner (point C). The distance from A to C is the diagonal of the square field.

### Step 2: Set Up the Problem
Let the side length of the square field be \( x \). The diagonal \( AC \) can be calculated using the Pythagorean theorem:
\[ AC = \sqrt{x^2 + x^2} = x\sqrt{2} \]

### Step 3: Use the First Angle
When the man is at point A, he observes the angle subtended by the pole at point C to be \( 60^\circ \). We can set up the following relationship using the tangent function:
\[ \tan(60^\circ) = \frac{h}{d} \]
where \( h \) is the height of the pole and \( d \) is the distance from A to the base of the pole (which is the diagonal \( AC \)):
\[ \sqrt{2}x = d \]
Thus, we have:
\[ \sqrt{3} = \frac{h}{x\sqrt{2}} \implies h = \sqrt{3} \cdot x\sqrt{2} = x\sqrt{6} \]

### Step 4: Use the Second Angle
When the man retreats 80 m from point A along the same line, he finds the angle to be \( 30^\circ \). Now, the distance from this new position (let's call it point B) to the pole is \( d + 80 \):
\[ \tan(30^\circ) = \frac{h}{d + 80} \]
Substituting \( h \) and \( d \):
\[ \frac{1}{\sqrt{3}} = \frac{x\sqrt{6}}{x\sqrt{2} + 80} \]

### Step 5: Cross-Multiply and Solve for \( x \)
Cross-multiplying gives:
\[ x\sqrt{2} + 80 = x\sqrt{6}\sqrt{3} \]
This simplifies to:
\[ x\sqrt{2} + 80 = 3x \]
Rearranging gives:
\[ 80 = 3x - x\sqrt{2} \]
Factoring out \( x \):
\[ 80 = x(3 - \sqrt{2}) \]
Thus:
\[ x = \frac{80}{3 - \sqrt{2}} \]

### Step 6: Rationalize the Denominator
To simplify \( x \):
\[ x = \frac{80(3 + \sqrt{2})}{(3 - \sqrt{2})(3 + \sqrt{2})} = \frac{80(3 + \sqrt{2})}{9 - 2} = \frac{80(3 + \sqrt{2})}{7} \]

### Step 7: Calculate the Length of the Field
Now, we can calculate the length of the field:
\[ \text{Length of the field} = x = \frac{80(3 + \sqrt{2})}{7} \]

### Final Answer
The length of the football field is approximately:
\[ \text{Length} \approx 34.29 \text{ m} \text{ (after calculating the numerical value)} \]