$(50$ points$)$ Rectangular fins with $k=250\frac{W}{m}K$ and thickness $t=2mm$ and length $L=3$ cm are on a $2cm2cm$ chip base. The chip base is $5$ mm thick, is made of material with $k=2\mathrm{.}5\frac{W}{m}K$ and puts out $20$ W total power. Air at $25C$ flowing around the fins induces a heat transfer coefficient of $150\frac{W}{m^2}K.$ There are $5$ fins and $2$ mm spacings $($see picture below$).$ You may assume the back and sides of the chip are insulated.

A$.(40$ points$)$ Find the base temperature.

B $(10$ points$)$ If the maximum temperature in the chip cannot exceed $90C,$ will this design work? Prove it$.$