Chapter 2.6, Problem 66E

Airplane Trajectory An airplane is flying at a speed of 350 mi/h at an altitude of one mile. The plane passes directly above a radar station at time t = 0.

1. (a) Express the distance s (in miles) between the plane and the radar station as a function of the horizontal distance d (in miles) that the plane has flown.
2. (b) Express d as a function of the time t (in hours) that the plane has flown.
3. (c) Use composition to express s as a function of t.

To determine

The function s that shows the distance between the plane and the radar station as a function of the horizontal distance d.

Answer to Problem 66E

The value of the function s is $s\left(d\right)=\sqrt{1+d^2}$.

Explanation of Solution

Given:

The speed of flying airplane is $350\text{mi/h}$.

The airplane is flying at an altitude of 1 mile.

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

Formula used: Pythagoras Theorem for right angle triangle.

${\left(\text{hypotenuse}\right)}^2={\left(\text{perpendicular}\right)}^2+{\left(\text{base}\right)}^2$

Calculation:

From the given information the airplane is flying at a speed of $350\text{mi/h}$ at an altitude $1\text{mi}$ and the plane passes directly above the radar station at time $t=0$, therefore the given information forms a right angle triangle as shown below,

Figure (1)

Since the function s that shows the distance between the plane and the radar station as a function of the horizontal distance d, therefore use the Pythagoras Theorem to find the function s.

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

Thus, the value of the function s is $s\left(d\right)=\sqrt{1+d^2}$.

To determine

The function d that shows the distance at time t.

Answer to Problem 66E

The value of the function d is $d\left(t\right)=350t$.

Explanation of Solution

Given:

The speed of flying airplane is $350\text{mi/h}$.

Calculation:

Since the function d is the function that shows the distance at time t, therefore the function d is written as the product of speed of airplane and time t and the speed of airplane is $350\text{mi/h}$.

$\begin{array}{c}d\left(t\right)=\left(350\right)\left(t\right)\\ =350t\end{array}$

Thus, the value of the function d is $d\left(t\right)=350t$.

To determine

The function s as a function of t by composition of functions.

Answer to Problem 66E

The function s as a function of t is expressed as $\left(s\circ d\right)\left(t\right)=\sqrt{1+122500t^2}$.

Explanation of Solution

Given:

From part (a), the value of function s is given below,

$s\left(d\right)=\sqrt{1+d^2}$ (1)

From part (b), the value of function d is given below,

$d\left(t\right)=350t$ (2)

Calculation:

The composite function $s\circ d$ is expressed as,

$\left(s\circ d\right)\left(t\right)=s\left(d\left(t\right)\right)$

From equation (2), substitute $350t$ for $d\left(t\right)$ in above expression,

$\left(s\circ d\right)\left(t\right)=s\left(350t\right)$ (3)

Substitute $350t$ for d in equation (1), to find the value of $s\left(350t\right)$,

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

Substitute $\sqrt{1+122500t^2}$ for $s\left(350t\right)$ in equation (3), to find the value of $s\circ d$,

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

The composite function $\left(s\circ d\left(t\right)\right)$ represents the function s as the function of time t.

Thus, the function s as a function of t is expressed as $\left(s\circ d\right)\left(t\right)=\sqrt{1+122500t^2}$.