a barn has a gable roof whose rafters are 20 feet long. the width of the barn is 30 feet. find the angle that the rafters make with the horizontal. find the area of one of the gable ends.
the angle is x, where cos x = 15/20
the area is 15√(20^2-15^2)
To find the angle that the rafters make with the horizontal, we can use trigonometry.
The width of the barn is 30 feet, and the length of the rafters is 20 feet. We can consider the triangle formed by the horizontal (base of the barn), the rafter (hypotenuse), and a vertical line (height of the barn).
Using the Pythagorean theorem, we can find the height of the barn:
height^2 = rafter^2 - base^2
height^2 = 20^2 - 30^2
height^2 = 400 - 900
height^2 = -500 (Since a negative height is not possible, we made an error in the calculation. Assuming there was a typo, let's calculate again using 20^2 - 15^2)
height^2 = 400 - 225
height^2 = 175
height = √175
height ≈ 13.23 feet
Now, we can calculate the angle that the rafters make with the horizontal using the tangent function:
tangent(angle) = height / base
tangent(angle) = 13.23 / 30
angle = arctan(13.23 / 30)
Using a scientific or graphing calculator, we can find the inverse tangent of (13.23 / 30) to determine the angle:
angle ≈ 24.17 degrees
Therefore, the angle that the rafters make with the horizontal is approximately 24.17 degrees.
To find the area of one of the gable ends, we need to calculate the area of a triangle. The formula for the area of a triangle is:
Area = (base * height) / 2
The base of the triangle is the width of the barn, which is 30 feet, and the height is the height we calculated earlier, which is approximately 13.23 feet.
Plugging these values into the formula, we get:
Area = (30 * 13.23) / 2
Area = 396.9 / 2
Area = 198.45 square feet
Therefore, the area of one of the gable ends is approximately 198.45 square feet.