Question

) An outdoor staircase has a total vertical elevation of 21 feet from the road level to the first floor. The 1st floor to the 2nd floor is also 21 feet. If the angle of elevation is 32 degrees and the landing of each step is 12 inches, answer the following questions showing each step of your work:

a) What is the vertical rise per step (in inches, rounded to the nearest inch)?


b) Find the number of steps needed to climb from road level to the 2nd floor.

Answer 1

a) 7 in

b) 72 steps

Explanation:

Angle of elevation formula:
`tan(\theta)=(y)/(x)`

where
`\theta` is the angle of elevation, y is the vertical distance (rise) and x is the horizontal distance.

a) Given:

• 
  `x=12`
• 
  `\theta=32`

`\implies tan(32)=(y)/(12)`

`\implies y=12tan(32)`

`\implies y=7 \ \textsf{in (nearest inch)}`

b) Given:

• 1st floor = 21 feet
• 2nd floor = 21 feet

⇒ Total height = 21 + 21 = 42 feet

1 foot = 12 inches

⇒ 42 feet = 42 x 12 = 504 inches

Using vertical rise of 7 in per step found in part (a):

Number of steps = 504 ÷ 7 = 72

Increased accuracy

Using unrounded vertical rise of 12tan(32):

Number of steps = 504 ÷ 12tan(32) = 67 steps to the nearest step