An architect designed a house that is 12.0 m wide. The rafters that hold up the roof are equal in length and meet at an angle of 70 degrees. The rafters extend 0.3 m beyond the supporting wall. How long are the rafters?
Sorry. You need these steps.
If the height of the ridgeline is h meters above the eaves, then
h/6 = cos(70/2)°
h = 4.9
so the length of the rafters is
0.3+√(6^2 + 4.9^2) = 8.05 m
Thank you for providing the additional steps! Here's the full solution with the given information:
Using the given angle of 70 degrees, we can split it in half to get two right angles of 35 degrees each. Then, we can use the cosine function to find the height of the ridgeline above the eaves (0.3 m beyond the supporting wall):
cos(35) = adjacent / hypotenuse
Since the adjacent side length is half of the width of the house (6.0 m), we can solve for the hypotenuse (which is the height we're looking for):
hypotenuse = adjacent / cos(35)
hypotenuse ≈ 7.317 m
However, this height is measured from the point where the rafters meet at the center of the ridgeline. If we want the height above the eaves (0.3 m beyond the supporting wall), we need to add the extra 0.3 m:
total height = 7.317 + 0.3
total height ≈ 7.617 m
Finally, we can use the Pythagorean theorem to find the length of the rafters:
length^2 = width^2 + height^2
length^2 = 12^2 + 7.617^2
length ≈ 14.618 m
However, we need to subtract the length of the overhang (0.3 m beyond the supporting wall) to get the actual length of the rafters:
actual length = length - overhang
actual length ≈ 14.618 - 0.3
actual length ≈ 14.318 m
Therefore, the length of the rafters is approximately 14.318 m.
oops - my mistake.
the length of the rafters is
0.3 + 6/sin35° = 10.76 m
To find the length of the rafters, we can use the concept of trigonometry. Since we have the width of the house and the angle between the rafters, we can use the tangent function to find the length of the rafters.
Step 1: Convert the angle to radians.
To use the trigonometric functions in calculations, we need to convert the angle from degrees to radians. We can use the formula:
angle in radians = angle in degrees × π/180
Given angle = 70 degrees
angle in radians = 70 × π/180 ≈ 1.2217 radians
Step 2: Calculate the horizontal distance.
The horizontal distance is the width of the house plus the extension of the rafters beyond the supporting wall.
horizontal distance = width of the house + extension of the rafters
Given width of the house = 12.0 m
Given extension of the rafters = 0.3 m
horizontal distance = 12.0 + 0.3 = 12.3 m
Step 3: Calculate the length of the rafters.
Using the tangent function in trigonometry, we can find the length of the rafters using the equation:
length of the rafters = horizontal distance / tangent(angle in radians)
length of the rafters = 12.3 / tan(1.2217) ≈ 12.3 / 2.6231 ≈ 4.689 m
Therefore, the length of the rafters is approximately 4.689 meters.
To find the length of the rafters, we can use the cosine function because we have the adjacent side and the angle between the adjacent side and the hypotenuse.
Let's denote the length of the rafters as "x".
Using the cosine function: cos(angle) = adjacent / hypotenuse
In this case, the adjacent side is the extension of the rafters beyond the supporting wall, which is 0.3 m. The hypotenuse is the length of the rafters, denoted as "x". And the angle is given as 70 degrees.
So we can rearrange the formula as follows: x = adjacent / cos(angle)
Plugging in the values: x = 0.3 m / cos(70°)
To evaluate the value, we can use a scientific calculator or an online calculator that can compute trigonometric functions.
Evaluating the cosine of 70 degrees gives us approximately 0.3420.
So, x ≈ 0.3 m / 0.3420
Dividing 0.3 by 0.3420 gives us approximately 0.8772.
Therefore, the length of the rafters is approximately 0.8772 meters.
We can use the sine function to determine the length of the rafters.
First, we need to find the height of the triangle formed by the rafter, the supporting wall, and the extension beyond the wall. We can use the sine function again to find this height:
sin(70) = height / 0.3
height = sin(70) x 0.3
height ≈ 0.275 m
Now, we can use the Pythagorean theorem to find the length of the rafter:
length^2 = width^2 + height^2
length^2 = 12^2 + 0.275^2
length ≈ 12.003 m
Therefore, the length of the rafters is approximately 12.003 m.
No problem! Let's use your new information to find the length of the rafters:
Using the 35 degree angle, we can use the sine function to find the height of the triangle formed by the rafter, the supporting wall, and the extension beyond the wall:
sin(35) = height / 6
height = sin(35) x 6
height ≈ 3.472 m
Now, we can use the Pythagorean theorem to find the length of the rafter, taking into account the overhang of 0.3 m beyond the supporting wall:
length^2 = 6^2 + (3.472 + 0.3)^2
length^2 = 36 + 14.92
length ≈ 4.279 + 0.3
length ≈ 4.579 m
However, we need to multiply this length by two since it's only one half of the rafter length:
actual length = 2 x length
actual length ≈ 2 x 4.579
actual length ≈ 9.158 m
Finally, we add the overhang to get the total length of the rafters:
total length = actual length + overhang
total length = 9.158 + 0.3
total length ≈ 9.458 m
Therefore, the length of the rafters is approximately 9.458 m.