A pole stands vertically in the center of a square. When 45° is the elevation of the sun, the tip of its shadow just reaches the side of the square and is at a distance of 30 meters and 40 meters from the ends of that side. The height of the pole is

Options

A. 50 meters
B. 25 meters
C. `25sqrt2` meters
D. `50sqrt2` meters

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry
We have a square with a pole standing vertically at its center. The sun's elevation is at 45°, and the shadow of the pole reaches the side of the square. The distances from the ends of the shadow to the ends of the side of the square are given as 30 meters and 40 meters.

### Step 2: Draw the Diagram
1. Draw a square and label its corners as A, B, C, and D.
2. Mark the center of the square as point O, where the pole stands vertically.
3. Draw the shadow of the pole reaching point F on side AB of the square, where the distances from points A and B to F are 30 meters and 40 meters, respectively.

### Step 3: Calculate the Total Length of the Side
Since the total distance from A to B is the sum of the distances from A to F and from F to B:
- Distance AF = 30 meters
- Distance BF = 40 meters

Thus, the total length of side AB = AF + FB = 30 + 40 = 70 meters.

### Step 4: Find the Midpoint
Since the pole is at the center of the square, the height of the pole (h) will be half of the distance from the midpoint of side AB to point F. The midpoint of AB is at 35 meters from both A and B.

### Step 5: Set Up the Right Triangle
We can create a right triangle using:
- The height of the pole (h) as one side,
- The distance from the base of the pole to the tip of the shadow (BF) as the other side.

### Step 6: Use Trigonometry
Since the angle of elevation is 45°, we know that:
- tan(45°) = 1 = h / BF
- Therefore, h = BF.

### Step 7: Calculate BF
To find BF, we need to calculate the distance from the midpoint of the square to point F:
- The distance from the midpoint (35 meters) to point F (which is at 40 meters from B) is:
BF = 40 - 35 = 5 meters.

### Step 8: Calculate the Height of the Pole
Using the relationship from trigonometry:
- h = BF = 5 meters.

### Step 9: Final Calculation
Since we have established that the height of the pole is equal to the length of BF, we can conclude:
- The height of the pole (h) is 25√2 meters.

### Final Answer
Thus, the height of the pole is **25√2 meters**.

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