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.
make a sketch by drawing in the height, calling it h
by Pythagoras:
h^2 + 15^2 = 20^2
h^2 = 175
h = √175
area = (1/2) base x height
= (1/2)(30)√175
=....
for angle:
tanØ = √175/15
I get Ø to be appr 41.4°
To find the angle that the rafters make with the horizontal, we can use trigonometry. Since we know the length of the rafters (20 feet) and the width of the barn (30 feet), we can use the tangent function to determine the angle.
The tangent of an angle is equal to the length of the side opposite the angle divided by the length of the side adjacent to the angle. In this case, the length of the side opposite the angle is 20 feet (the length of the rafters), and the length of the side adjacent to the angle is 30 feet (the width of the barn). So we can use the formula:
tan(angle) = opposite/adjacent
tan(angle) = 20/30
To find the angle, we can use the inverse tangent function (tan^-1) on both sides of the equation:
angle = tan^-1(20/30)
Using a scientific calculator or online calculator, we can find that the angle is approximately 33.69 degrees.
Now let's calculate the area of one of the gable ends. The gable end is a triangular shape formed by one side of the roof and the two sides of the barn.
To find the area of a triangle, we can use the formula:
Area = (1/2) * base * height
In this case, the base of the triangle is the width of the barn (30 feet), and the height is the length of the rafters (20 feet).
Area = (1/2) * 30 * 20
Area = 300 square feet
So the area of one of the gable ends is 300 square feet.