Given that the<span> 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 </span><span>risers and treads be n, then
12.94 < nx < 20 . . . (3)
and
ny = 11 . . . (4)
From (2), </span><span>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), </span><span>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
</span><span>2 < x + 2(0.55) < 2.08
2 < x + 1.1 < 2.08
0.9 < x < 0.98</span><span>
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.</span>