Now, the negative of this slope is
The reciprocal, then, is 24.
Again, let those who know correct me... ;-)
For (1)(a) the problem asked where the tangent line was perpendicular to y= 1- x/24. You found x where the derivative was 0. I have no idea why you did that! The derivative is the slope of the tangent line. A line with slope 0 is horizontal and is perpendicular to a vertical line. y= 1- x/24 is NOT vertical. It has slope -1/24. Clearly you are expected to know that if y= mx+ b and y= nx+ d are perpendicular, then n= -1/m. A line perpendicular to a line with slope -1/24 must have slope 24. THAT was why everyone set the derivative equal to 24, not 0!
For (1)(b), the problem asked where the tangent line was parallel to which has slope -12. Two lines are parallel if and only if they have the same slope.
Again- two lines, y= mx+ b and y= nx+ c, are parallel if and only if m= n, perpendicular if and only if m= -1/n (equivalently mn= -1).