A vertical stick 1.8 m long casts a shadow 45 cm long on the ground. At the same time, what is the length or the shadow of a pole 6 m high?

Options

A. `2.4m`
B. `1.35m`
C. `1.5m`
D. `13.5m`

Answer

To solve the problem of finding the length of the shadow of a pole that is 6 meters high, given that a vertical stick of 1.8 meters casts a shadow of 45 cm, we can use the concept of similar triangles. Here’s a step-by-step solution:

### Step 1: Understand the relationship between the heights and shadows
We have two vertical objects: a stick and a pole. The stick is 1.8 meters tall and casts a shadow of 45 cm. The pole is 6 meters tall, and we need to find the length of its shadow.

### Step 2: Convert all measurements to the same unit
To make calculations easier, we should convert all measurements to the same unit. Since the height of the stick is given in meters and the shadow in centimeters, we can convert the shadow length from centimeters to meters:
- 45 cm = 0.45 m

### Step 3: Set up the proportion using similar triangles
Since the two triangles formed by the stick and its shadow, and the pole and its shadow are similar, we can set up a proportion:
\[ \frac{\text{Height of the stick}}{\text{Length of the stick's shadow}} = \frac{\text{Height of the pole}}{\text{Length of the pole's shadow}} \]
Substituting the known values:
\[ \frac{1.8 \text{ m}}{0.45 \text{ m}} = \frac{6 \text{ m}}{x} \]
where \( x \) is the length of the shadow of the pole.

### Step 4: Cross-multiply to solve for \( x \)
Cross-multiplying gives us:
\[ 1.8 \cdot x = 6 \cdot 0.45 \]
Calculating the right side:
\[ 6 \cdot 0.45 = 2.7 \]
So we have:
\[ 1.8x = 2.7 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 1.8 to find \( x \):
\[ x = \frac{2.7}{1.8} \]
Calculating this gives:
\[ x = 1.5 \text{ m} \]

### Conclusion
The length of the shadow of the pole is **1.5 meters**.

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