A lean-to is a simple shelter with three walls, a sloping roof, and an open front facing away from the prevailing winds. The back wall is short compared to the front opening. If the lean-to at a campsite has a front opening that is 7.0 ft tall, a back wall that is 2.0 ft tall, and a floor that is 11 ft deep, what angle does the roofline make with the ground?
as always, draw a diagram. Since the front is 5' taller than the back, the angle you want is θ, such that
tanθ = 5/11
To find the angle that the roofline makes with the ground, we can use trigonometry.
First, let's define some variables:
- Height of the front opening (h): 7.0 ft
- Height of the back wall (h'): 2.0 ft
- Depth of the lean-to (d): 11 ft
We can consider the roofline as the hypotenuse of a right triangle, with the height difference between the front opening and the back wall as the vertical side, and the depth of the lean-to as the horizontal side.
Using the Pythagorean theorem, we can find the length of the roofline (hypotenuse):
roofline^2 = (d)^2 + (h - h')^2
roofline^2 = (11 ft)^2 + (7.0 ft - 2.0 ft)^2
roofline^2 = 121 ft^2 + 25 ft^2
roofline^2 = 146 ft^2
Next, let's find the angle. We can use the inverse tangent function (arctan) to calculate it:
angle = arctan((h - h') / d)
angle = arctan((7.0 ft - 2.0 ft) / 11 ft)
angle = arctan(5.0 ft / 11 ft)
Using a calculator, the approximate value of arctan(5.0 ft / 11 ft) is 24.41 degrees.
Therefore, the angle that the roofline makes with the ground is approximately 24.41 degrees.
To find the angle that the roofline makes with the ground, we can use trigonometry. Let's assume that the height of the front wall (H) is 7.0 ft, the height of the back wall (h) is 2.0 ft, and the depth of the lean-to (D) is 11 ft.
To find the angle, we need to calculate the tangent of the angle, which is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the difference in height between the front and back wall (H - h), and the adjacent side is the depth of the lean-to (D).
So, let's plug in the values into the tangent formula:
tan(θ) = (H - h) / D
tan(θ) = (7.0 - 2.0) / 11
tan(θ) = 5.0 / 11
Now, we need to take the inverse tangent (arctan) of both sides to find the angle θ:
θ = arctan(5.0 / 11)
Using a calculator, we find:
θ ≈ 24.9 degrees
Therefore, the angle that the roofline makes with the ground is approximately 24.9 degrees.