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