Chapter 1.3, Problem 56E

(a)

To determine

To express: The horizontal distance (in miles) covered by a plane as a function of time t.

Answer to Problem 56E

The function that represents the horizontal distance in terms of time t is $d\left(t\right)=350t$.

Explanation of Solution

Given:

The plane is flying at a speed of 350 mi/h and at an altitude of one mile.

The plane passes the radar station at time $t=0$.

Result used:

$\text{Distance= Time }\times \text{ Speed}$

Calculation:

Let the horizontal distance be d.

Substitute t for time and 350 for speed in the distance formula, $d\left(t\right)=350t$.

Thus, the function is $d\left(t\right)=350t$ where d(t) is measured in miles.

Therefore, the horizontal distance covered by the plane after t hours is $d\left(t\right)=350t$.

(b)

To determine

To express: The distance (s) between the plane and radar station in terms of d.

Answer to Problem 56E

The function that represents the distance between the plane and radar station in terms of the distance d is $s\left(d\right)=\sqrt{1+d^2}$.

Explanation of Solution

Let the horizontal distance be d in mi/hr.

It is given that the altitude is 1 mile.

Use the Pythagoras formula and obtain the value of s as follows.

$\begin{array}{c}{s}^2=1^2+d^2\\ \sqrt{s^2}=\sqrt{1^2+d^2}\\ s=\sqrt{1^2+d^2}\end{array}$

Thus, the function is $s\left(d\right)=\sqrt{1+d^2}$ where s(d) is measured in miles.

Therefore, the distance between the plane and radar station as a function of d is $s\left(d\right)=\sqrt{1+d^2}$.

(c)

To determine

To find: The expression for $\left(s\circ d\right)\left(t\right)$.

Answer to Problem 56E

The value of $\left(s\circ d\right)\left(t\right)=\sqrt{1+122500t^2}$.

Explanation of Solution

The composite function $\left(s\circ d\right)\left(t\right)$ is defined as $\left(s\circ d\right)\left(t\right)=s\left(d\left(t\right)\right)$.

From part (a), the value of $d\left(t\right)=350t$.

Thus, $\left(s\circ d\right)\left(t\right)=s\left(350t\right)$

Substitute $350t$ for d in $s\left(d\right)=\sqrt{1+d^2}$,

$\begin{array}{c}s\left(350t\right)=\sqrt{1+\left(350t^2\right)}\\ =\sqrt{1+122500t^2}\end{array}$

Thus, the required composition function, $\left(s\circ d\right)\left(t\right)=\sqrt{1+122500t^2}$.