Question $5$

$(i)$ Find the equation $of$ the line tangent $to$ the curve $xsin2y=ycos2x$

$at$ the point $(\frac{}{4},\frac{}{2}).$

$(ii)$ Let $abe$ a positive number, and let $y=$acosh$(\frac{x}{a}).$ Show how you may

obtain the differential equation $of$ the hanging cable, that $is,$

$\frac{d^{2}y}{d{x}^2}=\frac{1}{a}\sqrt[\phantom{2}]{1+(\frac{dy}{dx}{)}^2}$

$(iii)$ Show that the function $f_n(x,t)=sinnxcosnct$ satisfies the wave

equation $c^2\frac{de{l}^{2}f}{del{x}^2}=\frac{de{l}^{2}f}{del{t}^2},$ for any positive integer $n$ and any constant

$c.$

$(iv)$ A farmer has $600mof$ fencing with which $he$ plans $to$ add $a$

rectangular pen adjacent $to$ a long wall. $He$ will use the wall

$(which$ already exists$)$ for one side $of$ the pen and the fencing $to$

make the other three sides. What $is$ the maximum area $he$ can

enclose with the fencing $he$ has?