A man having height 169 cm is standing near a pole. He casts a shadow 130 cm long. What is the length of the pole if it gives a shadow 420 cm long?
169 cm लंबा एक आदमी एक खंभे के निकट खड़ा है। उसकी छाया 130cm लंबी पड़ती है। यदि खंभे की छाया 420 cm लंबी हो, तो उसकी लंबाई कितनी होगी?

Options

A. 546 cm
B. 550 cm
C. 589 cm
D. 323 cm

Explanation

To solve the problem of finding the height of the pole given the height of a man and the lengths of their shadows, we can use the concept of similar triangles. Here’s a step-by-step solution:

### Step 1: Understand the relationship between height and shadow length
The height of the man and the length of his shadow create a ratio that can be compared to the height of the pole and the length of its shadow.

### Step 2: Set up the known values
- Height of the man (HM) = 169 cm
- Length of the man's shadow (SM) = 130 cm
- Length of the pole's shadow (SP) = 420 cm
- Height of the pole (HP) = ?

### Step 3: Write the ratio of height to shadow length for the man
The ratio of the height of the man to the length of his shadow is:
\[ \frac{HM}{SM} = \frac{169}{130} \]

### Step 4: Write the ratio of height to shadow length for the pole
The ratio of the height of the pole to the length of its shadow is:
\[ \frac{HP}{SP} = \frac{HP}{420} \]

### Step 5: Set the two ratios equal to each other
Since the triangles are similar, we can set the ratios equal:
\[ \frac{169}{130} = \frac{HP}{420} \]

### Step 6: Cross-multiply to solve for HP
Cross-multiplying gives:
\[ 169 \times 420 = 130 \times HP \]

### Step 7: Calculate the left side
Calculating the left side:
\[ 169 \times 420 = 70980 \]

### Step 8: Solve for HP
Now, we can solve for HP:
\[ HP = \frac{70980}{130} \]

### Step 9: Perform the division
Calculating the division:
\[ HP = 546 \]

### Conclusion
The height of the pole is **546 cm**.

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