A vertical toy that is 18 cm long casts a shadow 8 cm long on level ground at a certain time of day. At the same time and in the same light, a pole casts a shadow 48 m long on the ground. Assuming both the toy and the pole are vertical, what is the height of the pole?

Introduction / Context:
This question is a classic application of similar triangles in height and distance problems. When the sun's rays fall at the same angle, any vertical objects and their shadows form similar right triangles. This means that the ratio of height to shadow length is the same for all such objects at that moment. Knowing the dimensions of a small object and its shadow allows us to scale up to find the height of a larger object from its shadow length.

Given Data / Assumptions:

• Height of the toy = 18 cm.
• Length of the toy's shadow = 8 cm.
• Length of the pole's shadow = 48 m.
• Both toy and pole are vertical and stand on level ground.
• Sun angle is the same for both, so the triangles formed are similar.
• We need to find the height of the pole.

Concept / Approach:
When two right triangles are similar, corresponding sides are proportional. Here, the toy and its shadow form a smaller right triangle, and the pole and its shadow form a larger, similar triangle. The ratio of height to shadow length for the toy equals the ratio of height to shadow length for the pole. We must also be careful with units: the toy measurements are in centimeters, while the pole's shadow is given in meters. It is helpful to work in consistent units, either all in centimeters or all in meters.

Step-by-Step Solution:
Step 1: Express the ratio of height to shadow for the toy. Toy height = 18 cm, toy shadow = 8 cm. Ratio = 18 / 8 = 9 / 4. Step 2: Convert the pole's shadow length to a compatible unit. Pole shadow = 48 m. Since 1 m = 100 cm, 48 m = 4800 cm. Step 3: Use similarity of triangles. Let H be the height of the pole in centimeters. Then H / 4800 = 18 / 8 = 9 / 4. Step 4: Solve for H. H = 4800 * (9 / 4) = 4800 * 9 / 4. First, 4800 / 4 = 1200. Then 1200 * 9 = 10800 cm. Step 5: Convert the pole height back to meters. 10800 cm = 10800 / 100 m = 108 m. Therefore, the height of the pole is 108 m.

Verification / Alternative check:
We can quickly check the proportionality. For the toy, height to shadow ratio is 18 : 8 = 9 : 4. For the pole, with height 108 m and shadow 48 m, the ratio is 108 : 48. Simplify by dividing both by 12: 108 / 12 = 9 and 48 / 12 = 4, so we also get 9 : 4. Because both ratios match, the triangles are indeed similar, and our scaled height is consistent with the geometry of the problem. This confirms that 108 m is correct.

Why Other Options Are Wrong:
1080 cm: This equals 10.8 m, which is ten times smaller than the correct height and does not maintain the 9 : 4 ratio with a 48 m shadow.
180 m: With a 48 m shadow, this would give a ratio of 180 : 48, which simplifies to 15 : 4, not 9 : 4, so it is inconsistent with the toy's ratio.
118 cm: This is even shorter than the toy in terms of centimeters and clearly does not scale correctly to a 48 m shadow, so this option is unrealistic here.
10.8 m: This is the same as 1080 cm, again too small and does not preserve the ratio of similar triangles. It seems to result from an incorrect scaling by a factor of 10 instead of 100.

Common Pitfalls:
The most common mistake in such questions is unit confusion. Students may try to directly use 48 along with 18 and 8 without converting meters to centimeters, causing a wrong proportion. Another frequent error is to invert the ratio, using shadow over height in one case and height over shadow in the other case, which breaks the similarity relationship. Keeping units consistent and writing down the proportional equation carefully prevents these mistakes.

Final Answer:
The height of the pole is 108 m.