A construction crew wants to hoist a heavy
beam so that it is standing up straight. Th ey
tie a rope to the beam, secure the base, and
pull the rope through a pulley to raise one
end of the beam from the ground. When
the beam makes an angle of 408 with the
ground, the top of the beam is 8 ft above
the ground.
Th e construction site has some telephone
wires crossing it. Th e workers are
concerned that the beam may hit the wires.
When the beam makes an angle of 608 with
the ground, the wires are 2 ft above the top
of the beam. Will the beam clear the wires
on its way to standing up straight?
To determine whether the beam will clear the wires, we need to calculate the height of the beam at an angle of 60° with the ground. We can use trigonometry to solve for the height.
Let's define a right-angled triangle with one leg representing the height of the beam (h), the other leg representing the distance between the base of the beam and the point on the ground directly below the top of the beam (x), and the hypotenuse representing the length of the beam (8 ft).
Using the sine function, we can write:
sin(60°) = h / 8
Rearranging the equation, we get:
h = 8 * sin(60°)
h ≈ 6.93 ft
Therefore, at an angle of 60°, the top of the beam is approximately 6.93 ft above the ground.
Now, let's determine whether the beam will clear the wires. We know that the wires are 2 ft above the top of the beam. So, the total height that the beam needs to clear is:
2 ft (height of wires) + 6.93 ft (height of the beam) = 8.93 ft
Since the beam is originally 8 ft long and the height it needs to clear is 8.93 ft, it is not possible for the beam to stand up straight without hitting the wires.