Problem #$5$: A hot drink is taken outside on a cold winter day when the air temperature is $-8C.$ According to a principle of physics called Newton's Law of Cooling, the temperature $T($in degrees Celsius$)$ of the drink $t$ minutes after being taken outside is given by

$T(t)=-8+A{e}^{-kt}$

where A and $k$ are constants.

$($a$)$ Suppose that the temperature of the drink is $88C$ when it is taken outside. Find the value of the constant $A.$

$($b$)$ In addition, suppose that $20$ minutes later the drink is $25C.$ Find the value of the constant $k.$

$($c$)$ What will the temperature be after $32$ minutes?

$($d$)$ When $($i$.$e$.,$ after how many minutes$)$ will the temperature reach $0C?$

Problem #$5($a$)$: $$

Problem #$5($b$)$: $$ Round your answer to $6$ decimals.

Problem #$5($c$)$: $$ Round your answer to $2$ decimals.

Problem #$5($d$)$: $$ Round your answer to $2$ decimals.