A vertical pole 2 m 80 cm high, casts a shadow 1 m 60 cm long. Find, at the same time
The height of the pole which casts a shadow of length 2 m 50 cm.

Answer

To solve the problem step by step, we will use the concept of direct variation.

### Step 1: Convert Measurements to Centimeters
First, we need to convert the heights and lengths of shadows from meters and centimeters to centimeters for consistency.

- Height of the pole = 2 m 80 cm = 280 cm (since 2 m = 200 cm and 80 cm = 80 cm, thus 200 cm + 80 cm = 280 cm).
- Length of the shadow = 1 m 60 cm = 160 cm (since 1 m = 100 cm and 60 cm = 60 cm, thus 100 cm + 60 cm = 160 cm).

### Step 2: Set Up the Proportion
According to the problem, the height of the pole and the length of the shadow are directly proportional. Therefore, we can set up the proportion as follows:

\[ \frac{\text{Height of first pole}}{\text{Length of first shadow}} = \frac{\text{Height of second pole}}{\text{Length of second shadow}} \]

Substituting the values we have:

\[ \frac{280 \text{ cm}}{160 \text{ cm}} = \frac{x}{250 \text{ cm}} \]

Where \( x \) is the height of the second pole that casts a shadow of 2 m 50 cm (which is 250 cm).

### Step 3: Cross-Multiply
Now, we will cross-multiply to solve for \( x \):

\[ 280 \times 250 = 160 \times x \]

This simplifies to:

\[ 70000 = 160x \]

### Step 4: Solve for \( x \)
To find \( x \), divide both sides by 160:

\[ x = \frac{70000}{160} \]

Calculating this gives:

\[ x = 437.5 \text{ cm} \]

### Step 5: Convert Back to Meters
Finally, we need to convert \( x \) back to meters:

\[ x = 437.5 \text{ cm} = 4.375 \text{ m} \quad (\text{since } 1 \text{ m} = 100 \text{ cm}) \]

### Final Answer
The height of the pole which casts a shadow of length 2 m 50 cm is **4.375 meters**.

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