College Algebra, Chapter 3, 3.6, Section 3.6, Problem 66

Suppose that an airplane is flying at a speed of $\displaystyle 350 \frac{\text{mi}}{\text{h}}$ 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.
Using Pythagorean Theorem,
$s(d) = \sqrt{d^2 + 1}$

b.) Express $d$ as a function of the time $t$ (in hours) that the plane has flown.
Recall that the formula for speed is $\displaystyle V = \frac{d}{t}$

$
\begin{equation}
\begin{aligned}
d &= Vt\\
\\
d &= 350 t
\end{aligned}
\end{equation}
$


c.) Use composition to express $s$ as a function of $t$.
Substitute value in part(b) using part(a), So

$
\begin{equation}
\begin{aligned}
s(350t) &= \sqrt{(350t)^2 + 1}\\
\\
s(350t) &= \sqrt{122500 t^2 + 1}
\end{aligned}
\end{equation}
$