The exact value of Cos(x/2) is -√(6-√31)/6.
The height of the bin B is 43.60 ft
How to determine half angle using half angle identity.
Given that sinx = √5/6 and π < x < 3π/2
cosx = +-√(1 - sin²x)
= +-√(1 - (√5/6)²)
= +-√{1 -5/36)
= +-√(31/36)
= +-√31/6
Cosx is negative in the third quadrant
cosx = -√31/6
Cos(x/2) = +-√(1 + cosx/2)
= +-√1 + (-√31/6)/2
= +-√(1 - √31/6)/2
= +-√(6 -√31/6)/2
= +-√(6 - √31)/3
Cos(x/2) = -√(6-√31)/6
2) tanα = 35/12
α = tan⁻¹(35/12)
= 71.06⁰
The angle formed by the auger and the ground for bin B
θ = α/2( it is half α)
The height of bin B using the formula:
h = 75 * sin(θ)
h = 75 * sin(α/2)
θ = α/2 ≈ 71.06/2 ≈ 35.54 degrees
Calculate h:
h = 75 * sin(θ)
h = 75 * sin(35.54)
= 75 * 0.5813
= 43.60 ft
Therefore, the height of the bin B is 43.60 ft.
complete question
Farming. An auger used to deliver grain to a storage bin can be raised and lowered, thus allowing for different size bins. Let α be the angle formed by the auger and the ground for bin A such that tanα = 35/12. The angle formed by the auger and the ground for bin B is
half of α. If the height h, in feet, of a bin can be found using the formula h = 75 sinθ, where θ is the angle formed by the ground and the
auger, find the height of bin B.