Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 55

Suppose that a ship is moving 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) We need to express the distance $s$ between the lighthouse and the ship as a function of $d$, the distance the ship has traveled since noon. Find $f$ so that $s = f(d)$.








by phytagorean theorem:


$
\begin{equation}
\begin{aligned}
s^2 =& d^2+6^2\\
s =& \sqrt{d^2+36}
\end{aligned}
\end{equation}
$



(b) We need to express $d$ as a function of $t$, the time elapsed since noon. Find $g$ so that $d = g(t)$.



$
\begin{equation}
\begin{aligned}
d = 30t
\end{aligned}
\end{equation}
$




(c) We need to find $f \circ g$ to know what this function represent.



$
\begin{equation}
\begin{aligned}

s =& \sqrt{d^2+36}; && d=30t\\
s =& \sqrt{(30t)^2+36}\\
s =& \sqrt{900t^2+36}

\end{aligned}
\end{equation}
$




It represents the distance between the lighthouse and the ship as a function ofthe time elapsed since noon.