$1.$ a$.$ Ice thickens over time $\backslash ($ t $\backslash )$ at a rate that is inversely proportional to its thickness $\backslash ($ T $\backslash )$ and directly proportional to the number of degrees Fahrenheit the temperature $\backslash ($ F $\backslash )$ is below $\backslash (32^\{\backslash$circ$\}\backslash )$ Fahrenheit. Write a differential equation to represent the rate of change in the thickness of the ice as time $\backslash ($ t $\backslash )$ passes at the same outside temperature $\backslash ($ F $\backslash ).$ Note that $\backslash ($ F $\backslash )$ can be thought of as a constant parameter, and note also that you will only need a single constant of proportionality in this differential equation.

b$.$ The rate at which a chapter of text is memorized is assumed to be proportional to the amount that is left to be memorized. Let $\backslash ($ M$($t$)\backslash )$ represent the amount of the chapter that is memorized as the decimal form of a percent, with a value between $\backslash (0=0\backslash \%\backslash )($none of it is memorized$)$ and $\backslash (1=100\backslash \%\backslash )($the chapter is completely memorized$).$ Write a differential equation to represent this situation.

c$.$ Let's extend the differential equation you developed for part b$.$ If the amount of the chapter that is forgotten in time $\backslash ($ t $\backslash )$ is proportional to the amount of the chapter memorized, $\backslash ($ M$($t$)\backslash ),$ write a new differential equation that takes this into account to represent the rate at which this chapter is memorized.