Your job is to design a staircase that has a total rise of 11 feet and meets these design guidelines:
1) the slope of the staircase is between 0.55 and 0.85
2) twice the rise plus the run must be between 24 and 25 inches
3) there can be no irregular steps
4) each step must be the same size
how many risers and treads?
what size are they?
explain design and include calculations.
show how each design guideline is met.

Given that the total rise of the staircase is 11 feet and the slope of the staircase is between 0.55 and 0.85

Then the total run of the stair case is between
. . . (1)

Given that twice the rise plus the run must be between 24 and 25 inches

Let the size of each run be x and the size of each rise, y, then
24 / 12 < x + 2y < 25 / 12

2 < x + 2y < 2.08 . . . (2)

Also, let the number of risers and treads be n, then
12.94 < nx < 20 . . . (3)
and
ny = 11 . . . (4)

From (2), 2 < x + 2y < 2.08, thus, we have
2 - 2y < x < 2.08 - 2y . . . (5)

Multiplying through by n, we have:
2n - 2ny < nx < 2.08n - 2ny . . . (6)

From (4), ny = 11, so we have
2n - 2(11) < nx < 2.08n - 2(11)
2n - 22 < nx < 2.08n - 22 . . . (7)

Comparing (3) and (7), we have
2n - 22 = 12.94 or 2.08n - 22 = 20

2n = 12.94 + 22 = 34.94 or 2.08n = 20 + 22 = 42

n = 17.47 or 20.19

Thus n is approximately 17 or 20.

From (4), ny = 11, so we have
y = 11/17 or 11/20
y = 0.65 or 0.55

From (2), 2 < x + 2y < 2.08, so we have
2 < x + 2(0.65) < 2.08
2 < x + 1.3 < 2.08
0.7 < x < 0.78

or
2 < x + 2(0.55) < 2.08
2 < x + 1.1 < 2.08
0.9 < x < 0.98

Therefore, we can conclude that we will have 20 risers and treads with each riser measuring 6.6 inches and each tread measuring 11 inches.