A barn is 30ft. wide by 60ft long; the rafters make an angle of 40degrees with the horizontal. Find the area of each of the two gable ends and the area of the roof.
Please Include Solution And If Possible An Illustration. Thank You In Advance :)
at each end, you have a triangle with base 15, and a height 15tan40°=12.58
so, each end has an area of 1/2 * 15 * 12.58 = 94.35
each half of the roof is 60 x 15sec40° = 60x19.58 = 1174.87
To find the area of each gable end and the roof, we need to calculate the height of the barn and the length of the rafters.
To find the height of the barn, we can use the angle of the rafters and the width of the barn.
The height (h) can be found using the formula:
h = width of barn * sin(angle)
In this case, the width of the barn is 30ft and the angle is 40 degrees:
h = 30 * sin(40°)
Using a calculator, we can find the value of sin(40°) is approximately 0.643:
h = 30 * 0.643 = 19.29ft
So, the height of the barn is approximately 19.29ft.
Now, to find the length of the rafters, we can use the Pythagorean theorem. The length of the rafters (r) can be calculated using the formula:
r = sqrt(h^2 + length of barn^2)
Substituting the values we have:
r = sqrt((19.29)^2 + 60^2)
Using a calculator, we find that the value inside the square root is approximately 3641.05:
r = sqrt( 3641.05 )
r ≈ 60.34 ft
So, the length of each rafter is approximately 60.34 ft.
To find the area of each gable end, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
The base of the gable end is the width of the barn, which is 30 ft. The height of the gable end is the height of the barn, which is 19.29 ft.
Area of each gable end = (1/2) * 30 * 19.29 = 289.35 sq ft
Therefore, the area of each gable end is approximately 289.35 sq ft.
To find the area of the roof, we can calculate the area of the rectangle formed by the length of the barn and the length of the rafter, and then subtract the areas of the two gable ends.
Area of the roof = (length of barn) * (length of rafter) - (2 * Area of each gable end)
Area of the roof = 60 * 60.34 - (2 * 289.35)
Area of the roof ≈ 3614.84 sq ft
Therefore, the area of the roof is approximately 3614.84 sq ft.
Here is an illustration of the barn with the calculations described above:
```
h
_______________
| / /
| / /
| / /
| / r /
|/_______ /
30 ft
```
I hope this helps! Let me know if you have any further questions.
To find the areas of the gable ends and the roof of the barn, we can divide the problem into two parts: finding the area of each gable end and finding the area of the roof.
Let's start with finding the area of each gable end:
Step 1: Draw a diagram.
Draw a rectangle to represent the barn, with a width of 30ft and a length of 60ft. Label the width as W and the length as L.
Step 2: Calculate the height of the gable end.
The rafters make an angle of 40 degrees with the horizontal. Since this forms a right triangle with the height of the gable end, we can use trigonometry to find the height.
In a right triangle, we have:
sin(angle) = opposite/hypotenuse.
In this case, the opposite side is the height of the gable end (H) and the hypotenuse is the length of the barn (L).
So, sin(40 degrees) = H/L.
To find H, multiply both sides by L:
H = L * sin(40 degrees).
Step 3: Calculate the area of each gable end.
The area of a triangle is given by the formula:
Area = base * height * 0.5.
For each gable end, the base is the width of the barn (W) and the height is the calculated H from step 2.
So, the area of each gable end is:
Area = W * H * 0.5.
Step 4: Calculate the area of the roof.
To find the area of the roof, we need to calculate the area of a trapezoid. A trapezoid has two parallel bases, and the height is the same as the height of the gable end.
The two bases of the trapezoid are the width of the barn (W) and the calculated height of the gable end (H). The length of the trapezoid is the length of the barn (L).
The area of a trapezoid is given by the formula:
Area = (base1 + base2) * height * 0.5.
So, the area of the roof is:
Area = (W + W) * H * 0.5.
Now, let's calculate the values.
Given:
W = 30ft (width of the barn),
L = 60ft (length of the barn),
Angle = 40 degrees.
Step 2 (continued): Calculating the height of the gable end.
H = L * sin(40 degrees).
H = 60ft * sin(40 degrees).
H ≈ 38.638ft.
Step 3 (continued): Calculating the area of each gable end.
Area = W * H * 0.5.
Area = 30ft * 38.638ft * 0.5.
Area ≈ 578.07ft².
Step 4 (continued): Calculating the area of the roof.
Area = (W + W) * H * 0.5.
Area = (30ft + 30ft) * 38.638ft * 0.5.
Area ≈ 1159.14ft².
Therefore, the area of each gable end is approximately 578.07 square feet, and the area of the roof is approximately 1159.14 square feet.
I apologize for the absence of illustrations, as I am a text-based AI and cannot provide visual representations. However, you can visualize the barn and the different areas described above by drawing a rectangle and adding the triangles/trapezoids based on the calculations provided.