A carpenter is building a staircase using his knowledge of right angles. The steps have a horizontal landing and a vertical rise causing each step to be a right triangle cut out. Depending on the purpose, steps may be adjusted by the angle of elevation which determines the pitch or steepness of the climb. A carpenter must factor in an appropriate amount of room on the landing for a person to have room for their foot on the step. Using this information, calculate the following questions below. (You must show all work to receive full credit)
1) Outdoor steps generally require larger landings and lower pitches. A particular outdoor staircase climbs a total vertical elevation of 13 feet from the road level to the front entrance of a building. If the angle of elevation is to be 25 degrees and the landing of each step is to be 28 inches, calculate the following (you must show all work to receive full credit):
a) What is the vertical rise per step? (In inches rounded to the nearest whole inch)
b) Find the number of steps needed to climb from road level to the front entrance.
2) 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):
2) 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.
3) A third staircase in a swimming pool has a landing of 15 inches and a vertical rise of 12 inches per step. To the nearest degree, calculate the following measures (you must show all work to receive full credit):
a) What is the angle of elevation rounded to the nearest whole degree?
b) If the elevation of each step were changed to 10 inches and the landing became 7 inches,
what is the new angle of elevation for the staircase to the nearest whole degree?
some hints to get you going:
#1
tan 25° = .466
So, each step is .466 as high as it is wide.
#2
tan 36.8° = 3/4
#3
tanθ = 12/15 = 4/5
θ = 38.7°
arctan(10/7) = 55°
To answer these questions, we need to use some trigonometry principles related to right triangles. Specifically, we will be using ratios of the sides of the triangle to find the unknowns.
1)
a) To find the vertical rise per step, we can use the trigonometric ratio tangent. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the opposite side represents the vertical rise per step, and the adjacent side represents the horizontal landing.
We know that the angle of elevation is 25 degrees and the landing of each step is 28 inches. Let's represent the vertical rise per step as x inches.
Using the tangent of the angle, we have the equation:
tan(25°) = x / 28
To solve for x, we can multiply both sides of the equation by 28:
28 * tan(25°) = x
Using a scientific calculator, we find that tan(25°) is approximately 0.466307658.
Now, we can calculate x:
x = 28 * 0.466307658 ≈ 13.05 inches
Therefore, the vertical rise per step is approximately 13 inches.
b) To find the number of steps needed to climb from road level to the front entrance, we can divide the total vertical elevation (13 feet) by the vertical rise per step (13 inches).
First, we need to convert the total vertical elevation from feet to inches:
13 feet = 13 * 12 inches = 156 inches
Now, we can calculate the number of steps:
156 inches / 13 inches = 12 steps
Therefore, the number of steps needed to climb from road level to the front entrance is 12.
2)
a) To find the landing space per step, we can use the trigonometric ratio tangent again. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the opposite side represents the vertical rise per step, and the adjacent side represents the landing space.
We know that the angle of elevation is 36.8 degrees and there are 16 feet of horizontal space total for each of the landings combined. Let's represent the landing space per step as x inches.
Using the tangent of the angle, we have the equation:
tan(36.8°) = x / (16 * 12)
To solve for x, we can multiply both sides of the equation by (16 * 12):
(16 * 12) * tan(36.8°) = x
Using a scientific calculator, we find that tan(36.8°) is approximately 0.772005948.
Now, we can calculate x:
x = (16 * 12) * 0.772005948 ≈ 147.33 inches
Therefore, the landing space per step is approximately 147.33 inches.
b) To find the vertical rise from the 1st floor to the 2nd floor, we can use the trigonometric ratio sine. The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse. In this case, the opposite side represents the vertical rise per step, and the hypotenuse represents the distance from the 1st floor to the 2nd floor.
We know that the angle of elevation is 36.8 degrees and the vertical rise per step is 9 inches. Let's represent the vertical rise from the 1st floor to the 2nd floor as x feet.
Using the sine of the angle, we have the equation:
sin(36.8°) = 9 / (x * 12)
To solve for x, we can multiply both sides of the equation by (x * 12):
(x * 12) * sin(36.8°) = 9
Using a scientific calculator, we find that sin(36.8°) is approximately 0.602011902.
Now, we can calculate x:
x = 9 / (sin(36.8°) * 12) ≈ 3.72 feet
Therefore, the vertical rise from the 1st floor to the 2nd floor is approximately 3.72 feet.
3)
a) To find the angle of elevation rounded to the nearest whole degree, we can use the inverse trigonometric function arctan (also known as tan^(-1)). This function gives us the angle whose tangent is equal to a given ratio.
We know that the landing is 15 inches and the vertical rise per step is 12 inches. Let's represent the angle of elevation as x degrees.
Using the arctan function, we have the equation:
x = arctan(12 / 15)
Using a scientific calculator, we find that arctan(12 / 15) is approximately 38.65980826.
Therefore, the angle of elevation rounded to the nearest whole degree is 39 degrees.
b) If the elevation of each step were changed to 10 inches and the landing became 7 inches, we can follow the same steps as in question 3a to find the new angle of elevation.
Using the arctan function, we have the equation:
x = arctan(10 / 7)
Using a scientific calculator, we find that arctan(10 / 7) is approximately 53.13010235.
Therefore, the new angle of elevation for the staircase, rounded to the nearest whole degree, is 53 degrees.