A vertical rod 20 m long casts a shadow 10 m long on the ground. At the same time a tower casts a shadow 50 m long on the same ground. The height of the tower is :

Question

Accepted Answer

To determine the height of the tower, we can use the concept of similar triangles. When the sun casts shadows, the triangles formed by the objects and their shadows are similar. This means that the ratios of the corresponding sides of these triangles are equal.

Given:
- A vertical rod is 20 m tall and casts a shadow 10 m long.
- A tower casts a shadow 50 m long.

We need to find the height of the tower.

Let's denote:
- Height of the rod = 20 m
- Shadow of the rod = 10 m
- Height of the tower = $ h $ m (unknown)
- Shadow of the tower = 50 m

Since the triangles are similar, the ratio of the height of the rod to its shadow is equal to the ratio of the height of the tower to its shadow:

$$
\frac{\text{Height of the rod}}{\text{Shadow of the rod}} = \frac{\text{Height of the tower}}{\text{Shadow of the tower}}
$$

Substituting the known values:

$$
\frac{20}{10} = \frac{h}{50}
$$

Simplifying the left side:

$$
2 = \frac{h}{50}
$$

To find $ h $, multiply both sides by 50:

$$
h = 2 \times 50 = 100
$$

Therefore, the height of the tower is 100 m, which corresponds to Option 1.

Explanation for why other options are incorrect:

- **Option 2 (120 m):** If the height of the tower were 120 m, the ratio would not match the ratio of the rod and its shadow. The calculation would be incorrect as it would not satisfy the proportion derived from similar triangles.
  
- **Option 3 (25 m):** A height of 25 m would result in a much smaller ratio, which does not align with the given shadow lengths and the height of the rod.

- **Option 4 (200 m):** A height of 200 m would result in a much larger ratio, which again does not match the proportion established by the rod and its shadow.

Thus, the only height that maintains the correct proportional relationship is 100 m, making Option 1 the correct answer.