Answer:
A) S = f(d) = √(36 + d²)
B) d = g(t) = 30t
C) f o g = √(36 + 900t²)
Explanation:
A) Express the distance s 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)
Refer to the attached image and observe that ship starts moving from point I towards point N which represents the position of ship at noon.
From point I to point N ship covers the distance d.
From point I to point L (lighthouse) the distance is S.
A triangle is formed and we want the distance S so we will apply the Pythagorean theorem
IL² = NL² + IN²
S² = 6² + d²
S = √(36 + d²)
Therefore, the function is S = f(d) = √(36 + d²)
B) Express d as a function of t, the time elapsed since noon;that is, find g, so that d = g(t)
Since the ship is moving at a speed of 30 km/h and let t represents the time taken to move from I to N then
d = 30t
or
d = g(t) = 30t
C) Find f o g. What does this function represent?
Since we have already found f(d) and g(t) we can find f o g.
f(d) = √(36 + d²) and d = g(t) = 30t
substitute d = g(t) = 30t into f(d)
f o g = √(36 + (g(t))²)
f o g = √(36 + (30t)²)
f o g = √(36 + 900t²)
This function simply represents the distance from point I to point N to point L which is the longer distance from the initial position of ship to the lighthouse.
Formula to find the arc length is:
Where, s= arc length,
r = radius of the circle
\theta = central angle in degrees.
According to the given problem, \theta= 150 and r =2.4.
So, first step is to plug in these values in the above formula to get the arc length.
=
So, arc length is .