Question

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 $t=0$.

(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.

(b) Express d as a function of the time t(in hours) that the plane has flown.

(c) Use composition to express s as a function of t.

Short answer

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

(b)The function das a function of the time t(in hours) that the plane has flown is$d\left(t\right)=350t$.

(c) The composition function is $\left(s\circ d\right)\left(t\right)=\sqrt{1+122500t^2}$.

Step-by-step solution

Part a. Step 1. The graphical interpretation of the data is shown below.

Given that 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 $t=0$.

Part a. Step 2. Pythagorean Theorem:

The Pythagorean Theorem states that for a right triangle, the square of the hypotenuse is sum of the squares of opposite side and adjacent side.

Part a. Step 3. Construct the function s:

The given triangle is a right triangle with length of hypotenuse is s, length of opposite is 1 mi, and the length of adjacent is d.

Now apply the Pythagorean Theorem as follows:

$$\begin{gathered} {\text{hypotenuse}}^2={\text{opposite}}^2+{\text{adjacent}}^2 \\ {s}^2=1^2+d^2 \\ s=\sqrt{1+d^2} \end{gathered}$$

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

Part b. Step 1. Definition of composition function.

Suppose$f\text{and}g$ are two functions, then composition function of$f\text{and}g$ is denoted as$f\circ g$ and defined as $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$.

Part b. Step 2. Definition of distance.

Let d be the distance travelled by an object, v is the speed of the object, and tis the time.

Then the distance formula is$d=vt.$

Part b. Step 3. Construct a function d.

The speed of the airplane is 350 mi/h.

So, the distance formula will be:

$$\begin{gathered} \text{distance}=\text{speed}\times \text{time} \\ d\left(t\right)=350t \end{gathered}$$

So, the function das a function of the time t(in hours) that the plane has flown is $d\left(t\right)=350t$.

Part c. Step 1. Definition of composition function.

Suppose$f\text{and}g$ are two functions, then composition function of$f\text{and}g$ is denoted$f\circ g$ as and defined as $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$.

Part c. Step 2. Recall the computed functions.

• 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 is$s\left(d\right)=\sqrt{1+d^2}$.
• The function das a function of the time t(in hours) that the plane has flown is $d\left(t\right)=350t$.

Part c. Step 3. Find s∘dt.

Use the composition definition and find$\left(s\circ d\right)\left(t\right):$

$$\begin{gathered} \left(s\circ d\right)\left(t\right)=s\left(d\left(t\right)\right)\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Definitionof}s\circ d \\ =s\left(350t\right)\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Definitionof}d \\ =\sqrt{1+{\left(350t\right)}^2}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}Definitionof}s \\ =\sqrt{1+122500t^2} \end{gathered}$$

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