Problem #$4$: The temperature at a point $(x,y)$ on a rectangular metal plate is given by

$T(x,y)=\frac{100}{4x^2+y^2}$

$($a$)$ Find the rate of change of $T$ at the point $(5,8)$ in the direction of the vector $(-3,-4)$

$($b$)$ At the point $(5,8),$ find a unit vector in the direction in which the temperature increases the fastest.

$($c$)$ What is the maximum rate of change of $T$ at the point $(5,8)?$

Problem #$4($a$)$: $$ Round your answer to $4$ decimals.

$q,$ Enter your answer $-\frac{4000}{?}$ sqrt $((-\frac{4000}{13456}{)}^2)$ symbolically, as in Enter the these examples

Problem #$4($b$)$:

$-\frac{4000}{\sqrt[\phantom{2}]{(\frac{-4000}{13456}{)}^2+(\frac{-1600}{13456}{)}^2}},-\frac{1600}{\sqrt[\phantom{2}]{(\frac{-4000}{13456}{)}^2+(\frac{-1600}{13456}{)}^2}}$ unit vector separated with a comma.

Problem #$4($c$)$: $$ Round your answer to $4$ decimals. Your work has been saved! $($Back to Admin Page$)$

Submit Problem #$4$ for Grading

$\backslash$table$[[,$Attempt #$2,$Attempt #$3,$Attempt #$4,$Attempt #$5],[500,4($a$)0\mathrm{.}18171600,4($a$)0,0773,4($a$),4($a$)],[60000,4($b$)\sqrt[\phantom{2}]{18560000},-\frac{1600}{\sqrt[\phantom{2}]{18560000}},4($b$)-\frac{4000}{67820},-\frac{1600}{67280},4($b$),4($b$)],[4($c$)4310\mathrm{.}4235,4($c$)4310\mathrm{.}4,4($c$),4($c$)],[4($a$)\frac{0}{2}x,4($a$)0/2$x$,4($a$),4($a$)],[4($b$)\frac{1\mathrm{.}5}{2}vx,4($b$)\frac{0}{2}x,4($b$),4($b$)],[4($c$)\frac{0}{2}x,4($c$)\frac{0}{2}x,4($c$),4($c$)]]$

Note: Your mark on each question will be the MAXIMUM of your marks on each try.

$($So there is no harm in making another attempt at a partially correct answer.$)$