When designing a stairway, an architect can use the riser-tread formula 2h + d = 25, where h is the riser height, in inches, and d is the tread depth, in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway. The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure above. The total rise of a stairway is the sum of the riser heights as shown in the figure. An architect wants to use the riser-tread formula to design a stairway with a total rise of 9 feet, a riser height between 7 and 8 inches, and an odd number of steps. With the architect’s constraints, which of the following must be the tread depth, in inches, of the stairway? (1 foot = 12 inches)

When designing a stairway, an architect can use the riser-tread form

1.	When designing a stairway, an architect can use the riser-tread formula 2h + d = 25, where h is the riser height, in inches, and d is the tread depth, in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway. The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure above. The total rise of a stairway is the sum of the riser heights as shown in the figure. An architect wants to use the riser-tread formula to design a stairway with a total rise of 9 feet, a riser height between 7 and 8 inches, and an odd number of steps. With the architect’s constraints, which of the following must be the tread depth, in inches, of the stairway? (1 foot = 12 inches)
Answer» 7.2 9.5 10.6 15 Solution :Let h be the riser height, in inches, and n be the number of the steps in the stairway. According to the architect’s design, the total RISE of the stairway is 9 feet, or 9 × 12 =108 inches. Hence, nh = 108, and solving for n gives `n=108/h`. It is given that 7 lt h lt 8. it FOLLOWS that `108/8 lt 108/h lt 108/7` , or equivalently , `108/8 lt n lt 108/7`. Since `108/8 lt 14` and `108/7 gt 15` and n is an integer , it follows that `14 le n le 15`.Since n can be an odd number, n can only be 15, therefore, `h=108/15=7.2` inches .Substituting 7.2 for h in the riser-tread formula 2H + d = 25 gives 14.4 + d =25. Solving for d gives d = 10.6 inches. Choice A is INCORRECT because 7.2 inches is the riser height, not the tread depth of the stairs. Choice B is incorrect and may be the result of CALCULATION errors. Choice D is incorrect because 15 is the number of steps, not the tread depth of the stairs.

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