Answer
The waterskier is rising at a rate of $~~7.73~ft/s$
Work Step by Step
Let $x$ be the horizontal length of the ramp and let $y$ be the height of the ramp. Let $L$ be the hypotenuse of the ramp.
We can find $L$ when $x = 15~ft$ and $y = 4~ft$:
$L^2 = x^2+y^2$
$L = \sqrt{x^2+y^2}$
$L = \sqrt{(15~ft)^2+(4~ft)^2}$
$L = \sqrt{241}~ft$
We can express $x$ in terms of $y$:
$x = \frac{15}{4}y$
We can find $\frac{dy}{dt}$:
$x^2+y^2 = L^2$
$(\frac{15}{4}y)^2+y^2 = L^2$
$\frac{225}{16}y^2+y^2 = L^2$
$\frac{241}{16}y^2 = L^2$
$\frac{241}{8}y~\frac{dy}{dt} = 2L~\frac{dL}{dt}$
$\frac{dy}{dt} = \frac{16L}{241y}~\frac{dL}{dt}$
$\frac{dy}{dt} = \frac{(16)(\sqrt{241}~ft)}{(241)(4~ft)}~(30~ft/s)$
$\frac{dy}{dt} = 7.73~ft/s$
The waterskier is rising at a rate of $~~7.73~ft/s$
