Luna is building a two piece roof for her shed. The width of the shed is 8ft, one roof piece is 4ft long and the other is 6ft long. What angle is formed at the peak of the roof?
Ans: 104 degrees ( I need the work)
divide all by two to make it easier
2,3,4
law of cosines
c^2 = a^2 + b^2 - 2 a b cos C
16 = 4 + 9 -2*2*3cos C
12 cos C = -3
cos C = -.25
C = 104.5 degees
To find the angle formed at the peak of the roof, we can use trigonometry. Let's assume that the longer roof piece is the base of a right triangle, the shorter roof piece is the height, and the angled line connecting the peak to the base is the hypotenuse.
Given:
Width of the shed = 8 ft
Length of one roof piece (base) = 6 ft
Length of the other roof piece (height) = 4 ft
To find the angle at the peak of the roof, we need to calculate the tangent of that angle:
tan(angle) = (height / base)
Substituting the given values:
tan(angle) = (4 / 6)
Now, we can use the inverse tangent function (arctan) to find the value of the angle:
angle = arctan(4 / 6)
Using a calculator, we can find the arctan(4 / 6) to be approximately 33.69 degrees.
However, this is the angle formed at one side of the roof's peak. To find the total angle formed at the peak, we need to double this value:
Total angle at the peak = 2 * 33.69 degrees
Calculating this, we get:
Total angle at the peak ≈ 67.38 degrees
Therefore, the angle formed at the peak of the roof is approximately 67.38 degrees.