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### Problem Breakdown:
- The ship moves parallel to the shore at **30 km/h**.
- At noon, the ship is **6 km from the shore**.
- There's a lighthouse on the shore, and the ship's movement is measured from the point it passes the lighthouse.

#### (a) Express \( s \) as a function of \( d \), where:
- \( s \) is the distance between the lighthouse and the ship.
- \( d \) is the distance the ship has traveled since noon.

We can use the Pythagorean theorem to express the relationship between the ship’s distance from the lighthouse and the shore:
- The ship's distance from the lighthouse (hypotenuse) is \( s \).
- The distance traveled parallel to the shore is \( d \).
- The perpendicular distance from the shore to the lighthouse is \( 6 \) km.

So, the relationship is:
\[
s = \sqrt{6^2 + d^2} = \sqrt{36 + d^2}
\]
Thus, the function \( s = f(d) \) is:
\[
s = f(d) = \sqrt{36 + d^2}
\]

#### (b) Express \( d \) as a function of \( t \), where:
- \( d \) is the distance traveled since noon.
- \( t \) is the time elapsed since noon.

Since the ship moves at 30 km/h, the distance it has traveled after \( t \) hours is:
\[
d = 30t
\]
Thus, the function \( d = g(t) \) is:
\[
d = g(t) = 30t
\]

#### (c) Find \( f \circ g \) and interpret what it represents.

The composition \( f \circ g \) means plugging \( g(t) \) into \( f(d) \), which gives us the distance between the lighthouse and the ship as a function of time \( t \).

So:
\[
f(g(t)) = f(30t) = \sqrt{36 + (30t)^2} = \sqrt{36 + 900t^2}
\]
Thus, \( f(g(t)) = \sqrt{36 + 900t^2} \).

This function represents the distance between the lighthouse and the ship at any time \( t \) after noon.

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Would you like more details or have any further questions? Here are some questions that expand on this:

1. How does the Pythagorean theorem apply in this scenario?
2. Why is the ship’s speed important for determining the function in part (b)?
3. What does the function \( f \circ g \) tell you about the ship's position over time?
4. How would the result change if the ship’s initial distance from the shore was different?
5. Can you graph the function \( f(g(t)) \) to visualize the relationship?

**Tip:** Understanding the composition of functions \( f \circ g \) helps in modeling real-world relationships, like the ship’s distance from the lighthouse over time.

A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon. (b) Express d as a function of t, the time elapsed since noon. (c) Find f ∘ g. What does this function represent?

Learn how to express the distance between a ship and a lighthouse using the Pythagorean theorem and function composition. This problem involves a ship moving at 30 km/h parallel to the shore, passing a lighthouse 6 km away at noon.

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