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 need to find the height of the pole based on the given information about the shadow and the square.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a square with a pole at its center. The sun's elevation is at 45°, which means the angle of elevation from the tip of the shadow to the top of the pole is 45°.
- Let’s denote the height of the pole as \( h \).

2. **Identifying the Shadow**:
- The tip of the shadow (point Q) is at a distance of 30 meters from one end of the side (point A) and 40 meters from the other end (point D).
- Therefore, the total length of the side of the square (AD) can be calculated as:
\[ AD = AQ + QD = 30 + 40 = 70 \text{ meters} \]

3. **Finding the Midpoint**:
- Since the pole is at the center of the square, the midpoint (M) of side AD is:
\[ AM = \frac{AD}{2} = \frac{70}{2} = 35 \text{ meters} \]

4. **Setting Up the Right Triangle**:
- We can form a right triangle \( OQM \) where:
- \( O \) is the top of the pole,
- \( Q \) is the tip of the shadow,
- \( M \) is the midpoint of side AD.
- The lengths we know are:
- \( QM = AM - AQ = 35 - 30 = 5 \text{ meters} \)
- \( OM = 35 \text{ meters} \) (since it is half the side of the square).

5. **Applying the Pythagorean Theorem**:
- In triangle \( OQM \):
\[ OQ^2 = OM^2 + QM^2 \]
- Substituting the known values:
\[ OQ^2 = 35^2 + 5^2 = 1225 + 25 = 1250 \]
- Therefore, the length of \( OQ \) is:
\[ OQ = \sqrt{1250} = 25\sqrt{2} \text{ meters} \]

6. **Finding the Height of the Pole**:
- Since the angle of elevation is 45°, we have:
\[ OP = OQ = h \]
- Thus, the height of the pole is:
\[ h = 25\sqrt{2} \text{ meters} \]

### Final Answer:
The height of the pole is \( 25\sqrt{2} \) meters.