A ship is moving at a speed of 15 km/h parallel to a straight shoreline. The ship is 9 km from shore, and it passes a lighthouse at noon.
(a)
Express the distance s (in km) between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that
s = f(d).
s = f(d) =
Since the ship is moving parallel to the shoreline, the distance between the ship and the lighthouse will remain constant. Therefore, the distance s between the lighthouse and the ship is equal to the initial distance of 9 km.
Thus, the function f(d) = 9 km.
To find the distance s between the lighthouse and the ship as a function of d, you can use the Pythagorean theorem.
Let's assume that the ship is traveling directly away from the lighthouse. In this case, the distance the ship has traveled since noon, d, is the hypotenuse of a right triangle, and the distance between the lighthouse and the ship, s, is one of the legs of the triangle.
Using the Pythagorean theorem, we can express s in terms of d:
s^2 = d^2 - 9^2
Taking the square root of both sides, we get:
s = sqrt(d^2 - 81)
Therefore, the function f(d) is:
f(d) = sqrt(d^2 - 81)