Indoor steps generally are steeper and have smaller landings. A staircase has a 9 inch vertical rise per step to get to the second floor of a house. If the angle of elevation is 36.8 degrees and there are 16 feet of horizontal space total for each of the landings combined, calculate the following questions (you must show all work to receive full credit):
a) What is the landing space per step? (In inches rounded to the nearest tenth)
b) Find the vertical rise from the 1st floor to the 2nd floor to the nearest foot.
To solve this problem, we need to use trigonometry and convert units appropriately. Let's go through the steps:
a) To find the landing space per step, we'll start by converting the total horizontal space for the landings from feet to inches. Since there are 12 inches in a foot, we'll multiply the total horizontal space by 12:
Total horizontal space = 16 feet * 12 inches/foot = 192 inches
Now we need to find the horizontal distance per step. Since there are n steps, we can divide the total horizontal space by the number of steps to get the horizontal space per step, which we'll call X:
X = 192 inches / n
Next, we'll use trigonometry to relate the angle of elevation and the vertical rise per step. The tangent of an angle is equal to the vertical rise divided by the horizontal run. In this case, the vertical rise is 9 inches and the horizontal run is X inches per step. So we have the following equation:
tan(36.8 degrees) = 9 inches / X inches
To solve for X, we can rearrange the equation:
X inches = 9 inches / tan(36.8 degrees)
Using a calculator, we find that tan(36.8 degrees) is approximately 0.7568. Therefore:
X inches = 9 inches / 0.7568 ≈ 11.9 inches
So the landing space per step is approximately 11.9 inches, rounded to the nearest tenth.
b) To find the vertical rise from the 1st floor to the 2nd floor, we can multiply the number of steps by the vertical rise per step:
Vertical rise = n steps * 9 inches/step
We don't have the exact number of steps (n), but we know the total horizontal space for the landings is 16 feet. Assuming an average stride length for a step is 2.5 feet, we can estimate the number of steps by dividing the total horizontal space by the stride length:
n ≈ 16 feet / 2.5 feet/step = 6.4 steps
Since we can't have decimal steps, we'll round down to 6 steps.
Vertical rise = 6 steps * 9 inches/step = 54 inches
Converting inches to feet, we have:
Vertical rise ≈ 54 inches / 12 inches/foot = 4.5 feet
So, the vertical rise from the 1st floor to the 2nd floor is approximately 4.5 feet, rounded to the nearest foot.