A fox, camouflaged in the bush, observes from the ground, that it takes 2 minutes for the angle of elevation of the squirrel to change from `30^@` to `60^@`. If the speed of fox is `sqrt3` times that of the squirrel, find the time taken by fox to reach the tree from the bush.

Options

A. 1 minute
B. `sqrt3` minute
C. 3 minute
D. `4/sqrt3` minute

Answer

To solve the problem, we will break it down into steps.

### Step 1: Understand the scenario
The fox observes the squirrel from the ground. The angle of elevation changes from \(30^\circ\) to \(60^\circ\) over a period of 2 minutes. We need to find the time taken by the fox to reach the tree where the squirrel is.

### Step 2: Set up the right triangles
Let:
- \(AB\) be the height of the tree (the vertical distance from the ground to the squirrel).
- \(BC\) be the horizontal distance from the fox to the base of the tree when the angle of elevation is \(30^\circ\).
- \(CE\) be the horizontal distance from the fox to the base of the tree when the angle of elevation is \(60^\circ\).

### Step 3: Use trigonometric ratios
For the first triangle \(ABC\) (when the angle of elevation is \(30^\circ\)):
\[ \tan(30^\circ) = \frac{AB}{BC} \implies \frac{1}{\sqrt{3}} = \frac{AB}{BC} \implies AB = \frac{BC}{\sqrt{3}} \quad \text{(1)} \]

For the second triangle \(DCE\) (when the angle of elevation is \(60^\circ\)):
\[ \tan(60^\circ) = \frac{DE}{CE} \implies \sqrt{3} = \frac{DE}{CE} \implies DE = \sqrt{3} \cdot CE \quad \text{(2)} \]
Since \(AB = DE\), we can equate (1) and (2):
\[ \frac{BC}{\sqrt{3}} = \sqrt{3} \cdot CE \]

### Step 4: Solve for the distances
Cross-multiplying gives:
\[ BC = 3 \cdot CE \quad \text{(3)} \]

### Step 5: Determine the distance traveled by the squirrel
The squirrel moves from point \(C\) to point \(E\) in 2 minutes. The distance \(CE\) is 1 unit (from the ratio derived earlier) and \(BC\) is 3 units. Therefore, the distance traveled by the squirrel is:
\[ BE = BC - CE = 3 - 1 = 2 \text{ units} \]

### Step 6: Calculate the speed of the squirrel
The speed of the squirrel is:
\[ \text{Speed of squirrel} = \frac{\text{Distance}}{\text{Time}} = \frac{2 \text{ units}}{2 \text{ minutes}} = 1 \text{ unit/minute} \]

### Step 7: Calculate the speed of the fox
The speed of the fox is given as \(\sqrt{3}\) times the speed of the squirrel:
\[ \text{Speed of fox} = \sqrt{3} \cdot 1 = \sqrt{3} \text{ units/minute} \]

### Step 8: Calculate the time taken by the fox to reach the tree
The distance the fox needs to cover is \(BC\) which is 3 units. The time taken by the fox is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{3 \text{ units}}{\sqrt{3} \text{ units/minute}} = \sqrt{3} \text{ minutes} \]

### Final Answer
The time taken by the fox to reach the tree from the bush is \(\sqrt{3}\) minutes.

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