A hot metal bar is submerged in a large reservoir of water whose temperature is $45^\circ \text{F}$. The temperature of the bar $20\text{s}$ after submersion is $85^\circ \text{F}$. After $1\,\text{min}$ submerged, the temperature has cooled to $65^\circ \text{F}$.

(a) Determine the cooling constant $k$.

$k =$ $s^{-1}$

(b) What is the differential equation satisfied by the temperature $F(t)$ of the bar?

$F'(t) =$ Use F for F(t).

(c) What is the formula for $F(t)$?

$F(t) =$

(d) Determine the temperature of the bar at the moment it is submerged.

Initial temperature = ${}^\circ \text{F}$