A flow rate sensing device used on a liquid transport pipeline functions as follows. The device provides a $5-$bit output where

all five bits are zero if the flow rate is less than $10$ gallons per minute. The first bit is $1$ if the flow rate is at least $10$ gallons

per minute; the first and second bits are $1$ if the flow rate is at least $20$ gallons per minute; the first, second, and third bits

are $1$ if the flow rate is at least $30$ gallons per minute; and so on$.$ The five bits, represented first through fifth by the logical

variables $A,B,C,D,$ and $E,$ respectively, are used as inputs to a device that provides two outputs $Y$ and $Z.($Use the

following as necessary: $A,B,C,D,$ and $E.)$

$($a$)$ Write an equation for the output $Y$ if we want $Y$ to be $1$ if the flow rate is less than $40$ gallons per minute.

$Y=$

$($b$)$ Write an equation for the output $Z$ if we want $Z$ to be $1$ if the flow rate is at least $10$ gallons per minute but less than

$40$ gallons per minute.