Question

A ship is moving at a speed of 30 km/h parallel to a straightshoreline. The ship is 6 km from the shore and it passes alighthouse at noon.
A) Express the distance s between the lighthouse and the shipas a function of d, the distance the ship has traveled since noon;that is, find f so that s = f(d).
B) Express d as a function of t, the time elapsed since noon;that is, fing g, so that d = g(t).
C) Find f o g. What does this functionrepresent?

Answer 1

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.