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).
Since the ship is moving parallel to the shoreline, the distance s between the lighthouse and the ship is the same as the distance the ship has traveled since noon.
Therefore, the function that expresses s as a function of d is:
s = d
To express the distance s between the lighthouse and the ship as a function of d, we can use the Pythagorean Theorem.
Let's define the following:
- d: the distance the ship has traveled since noon (in km)
- s: the distance between the lighthouse and the ship (in km)
- x: the distance the ship has traveled parallel to the shore (in km)
From the problem statement, we know that the ship is 9 km from the shore. Therefore, the distance traveled parallel to the shore is fixed at 9 km.
So we have:
x = 9 km
Since the speed of the ship is 15 km/h, the time t (in hours) since noon is given by:
t = d / 15
Now, we can use the Pythagorean Theorem to find s in terms of x and t:
s^2 = x^2 + (15t)^2
Substituting the values of x and t, we get:
s^2 = 9^2 + (15(d / 15))^2
Simplifying, we have:
s^2 = 81 + d^2
Taking the square root of both sides, we get:
s = โ(81 + d^2)
Therefore, the function f(d) that expresses the distance s between the lighthouse and the ship as a function of d is:
f(d) = โ(81 + d^2)