Chapter 1.3, Problem 55E

A ship is at a speed of 30km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse 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 = q(t).

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

To determine

To express: The distance between the lighthouse and the ship as a function of d.

Answer to Problem 55E

The distance between the light house and the ship as of function d is, $s=\sqrt{d^2+36}$.

Explanation of Solution

Given:

The speed of the ship is 30 km/h.

The distance between the ship and shore is 6 km.

Diagram:

Draw a rough diagram to indicate the positions of the ship, lighthouse and shore as shown below in Figure 1.

Consider the right triangle from Figure 1.

In order to find the distance s, use Pythogorean theorem.

$\begin{array}{l}{s}^2=d^2+6^2\\ s=\sqrt{d^2+36}\\ s=f\left(d\right)=\sqrt{d^2+36}\end{array}$

Therefore, the distance between the light house and the ship as a function d is, $s=\sqrt{d^2+36}$.

To determine

To express: The function d as a function of t where t is the time elapsed since noon.

Answer to Problem 55E

The function d as a function of t is, $d=30t$.

Explanation of Solution

Given:

The speed of the ship is 30 km/h.

d is the distance in terms of kilometer and t is the time in hours.

Distance travelled is the product of speed and time.

Thus, the required function is, $d=30t$.

To determine

To find: The composition function $f\circ g$ and explain.

Answer to Problem 55E

The composition function $f\circ g=6\sqrt{25t^2+1}$ and this function represents the distance between the light house and the ship since noon.

Explanation of Solution

From part(a), $s=f\left(d\right)=\sqrt{d^2+36}$ and from part b, $g\left(t\right)=d=30t$.

The composition function is, $f\circ g=f\left(g\left(t\right)\right)$.

Then $f\left(g\left(t\right)\right)=\sqrt{g{\left(t\right)}^2+36}$

Substitute $g\left(t\right)$ and find the value of the composition function.

$\begin{array}{c}f\left(30t\right)=\sqrt{{\left(30t\right)}^2+36}\\ =\sqrt{900t^2+36}\\ =\sqrt{36\times 25t^2+36}\\ =6\sqrt{25t^2+1}\end{array}$

Therefore, the composition function is, $f\circ g=6\sqrt{25t^2+1}$ and it represents the distance between the light house and the ship in terms of the elapsed time since noon.