Let's analyze and solve the given problem:

Problem Restatement:

• A lighthouse is located 10 m back from the edge of a 35 m high cliff.
• The beacon is 8 m above the base of the lighthouse (making the beacon's height from the cliff base $35 + 8 = 43$ m).
• We are tasked with finding how far from the cliff the shadow of the beacon extends onto the water.

Solution Approach:

The problem can be solved using similar triangles.

Step 1: Key Measurements

1. Height of the beacon from the cliff base = $43$ m.
2. Distance of the lighthouse from the edge of the cliff = $10$ m.

Step 2: Form Similar Triangles

The light forms a triangle:

1. One side is the vertical height from the cliff to the beacon $(43 \, \text{m})$,
2. The shadow forms the horizontal distance $x$, starting from the edge of the cliff.

For similar triangles: $\frac{\text{Height of beacon}}{\text{Distance from edge of cliff to shadow's end}} = \frac{\text{Height of cliff}}{\text{Distance of lighthouse from edge of cliff}}.$

Step 3: Write and Solve the Equation

Let the shadow extend $x$ meters from the edge of the cliff. Then: $\frac{43}{x} = \frac{35}{10}.$

Cross-multiply: $43 \cdot 10 = 35 \cdot x.$

Simplify: $430 = 35x.$

Solve for $x$: $x = \frac{430}{35} \approx 12.29 \, \text{m}.$

Final Answer:

The shadow extends approximately 12.29 m from the edge of the cliff.

Would you like a detailed diagram or additional clarification? Let me know!

Here are 5 related questions you may explore:

1. What is the total distance from the lighthouse to the tip of the shadow?
2. How would the result change if the cliff height increased?
3. What if the beacon height above the lighthouse was unknown—how could we estimate it?
4. Can you calculate the angle of elevation of the beacon from the shadow's tip?
5. What are the practical uses of similar triangles in engineering?

Tip: Always verify the proportionality of triangles when applying similar triangle rules to real-world problems.