Calculus, 8th Edition,stewart; page 44 problem 57

Question

A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a light house at noon.

(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).

(b) Express d as a function of t, the time elapsed since noon; that is, find g so that d = g (t).

(c) Find f o g. What does this function represent ?

Answer 1

Step:1

The ship is moving at a speed of  km/h.

The ship is travelling parallel to shoreline and at noon the ship passes the light house.

At noon, the ship is  km from shore.

(a)

Find .

Observe the above figure.

In the beginning, the ship is  distance away from the light house .

At noon time, the distance between ship and light house is  6 km.

 d is the distance traveled by the ship.

From Pythagorean theorem,

The function of distance traveled by the ship is .

Step:2

(b)

Find .

 is the distance as a function of time.

Let  be the time taken by the ship to travel a distance of .

The speed of the ship is 30 km/h.

Speed-distance relation:

.

Therefore .

.

The distance traveled by the ship since noon time is .

Step:3

(c)

Find .

                                               ( Since  )

             

.

It represents the distance between the ship and the light house as the function of time elapsed since noon.

Solution:

(a) .

(b) .

(c) , it represents the distance between the ship and the light house as the function of time elapsed since noon.