To express the distance \(s\) between the lighthouse and the ship as a function of \(d\), where \(d\) is the distance the ship has traveled since noon, you can use the Pythagorean theorem since the ship is moving parallel to the shoreline.
Let \(s\) be the distance between the lighthouse and the ship, and \(d\) be the distance the ship has traveled since noon. Also, let \(x\) be the distance the ship has traveled parallel to the shoreline.
According to the Pythagorean theorem:
\[s^2 = d^2 + x^2\]
We know that the ship is moving at a speed of 20 km/h, and it passes a lighthouse at noon. So, the distance \(x\) traveled parallel to the shoreline is the ship's speed multiplied by the time since noon, which is \(d\) hours.
\[x = 20d\]
Now, substitute this into the Pythagorean theorem:
\[s^2 = d^2 + (20d)^2\]
Simplify the equation:
\[s^2 = d^2 + 400d^2\]
Combine like terms:
\[s^2 = 401d^2\]
Now, take the square root of both sides to find \(s\) as a function of \(d\):
\[s = \sqrt{401d^2}\]
\[s = 20d\sqrt{401}\]
So, \(s = f(d) = 20d\sqrt{401}\) represents the distance between the lighthouse and the ship as a function of \(d\).