Two friends, Amy and Bob, are standing in line with a lamp post. The shadows of both friends meet at the same point on the ground. If the heights of the lamp post, Amy and Bob are 6 meters, 1.8 meters and 0.9 meters respectively, and Amy is standing 2 meters away from the post, then how far (in meters) is Bob standing from Amy?

Options

A. 0.43
B. 0.9
C. 1.8
D. 2

Answer

To solve the problem, we will use the concept of similar triangles formed by the lamp post and the shadows of Amy and Bob. Let's break down the solution step by step.

### Step 1: Understand the setup
We have a lamp post of height 6 meters, Amy with a height of 1.8 meters standing 2 meters away from the lamp post, and Bob with a height of 0.9 meters. The shadows of both Amy and Bob meet at the same point on the ground.

### Step 2: Draw the diagram
- Let \( A \) be the top of the lamp post (6 m).
- Let \( B \) be the point where Amy is standing (1.8 m) and is 2 m away from the lamp post.
- Let \( C \) be the point where Bob is standing (0.9 m) at a distance \( x \) from Amy.
- Let \( D \) be the point where the shadows meet on the ground.

### Step 3: Identify the triangles
We can identify two right triangles:
1. Triangle \( ADB \) (formed by the lamp post and Amy)
2. Triangle \( AEC \) (formed by the lamp post and Bob)

### Step 4: Calculate the height difference for Amy
The height from the top of the lamp post to the top of Amy's head is:
\[ AH = AB - HB = 6 - 1.8 = 4.2 \text{ meters} \]

### Step 5: Use the tangent ratio for Amy's triangle
Using the tangent of the angle formed by the lamp post and Amy's shadow:
\[ \tan(\theta) = \frac{AH}{GB} = \frac{4.2}{2} \]
Thus,
\[ \tan(\theta) = 2.1 \]

### Step 6: Calculate the height difference for Bob
The height from the top of the lamp post to the top of Bob's head is:
\[ GI = GC - IC = 1.8 - 0.9 = 0.9 \text{ meters} \]

### Step 7: Use the tangent ratio for Bob's triangle
Using the tangent of the angle formed by the lamp post and Bob's shadow:
\[ \tan(\theta) = \frac{GI}{FI} = \frac{0.9}{x} \]

### Step 8: Set the tangents equal
Since both triangles share the same angle \( \theta \):
\[ \frac{0.9}{x} = 2.1 \]

### Step 9: Solve for \( x \)
Cross-multiplying gives:
\[ 0.9 = 2.1x \]
Now, solving for \( x \):
\[ x = \frac{0.9}{2.1} = \frac{9}{21} = \frac{3}{7} \approx 0.42857 \text{ meters} \]

### Step 10: Round the answer
Rounding to two decimal places, Bob is approximately:
\[ x \approx 0.43 \text{ meters away from Amy} \]

### Summary
Bob is standing approximately 0.43 meters away from Amy.

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