A construction crew wants to hoist a heavy
beam so that it is standing up straight. They
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 40 degrees 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 60 degrees 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?
we see that the length of the beam is
8/sin40 = 12.45 ft
At 60 degrees, the top is
12.45sin60 = 10.78 ft high
So, the wire is 12.78 ft up.
Since the beam is only 12.45 ft long, it will not touch the wires.
Well, it sounds like we've got some high stakes acrobatics going on at this construction site! Let's see if our beam can clear those pesky wires.
First, let's break it down. When the beam is at a 40-degree angle with the ground and the top of the beam is 8 ft above the ground, we can call this distance "A."
Now, when the beam reaches a 60-degree angle with the ground and the wires are 2 ft above the top of the beam, we can call this distance "B."
To determine if the beam will clear the wires, we need to compare distances A and B. If A is greater than B, we're in the clear. If B is greater than A, we may have a shocking situation on our hands.
Now, here comes the fun part - math! We can use some good old trigonometry to solve for the two distances.
Using the sine function, we can set up the following equation:
sin(40 degrees) = A / 8 ft
Solving for A, we get:
A = 8 ft * sin(40 degrees)
Similarly, for distance B, we have:
sin(60 degrees) = B / (8 ft + 2 ft)
Solving for B, we get:
B = (8 ft + 2 ft) * sin(60 degrees)
Now, let's plug in those values and calculate A and B:
A = 8 ft * sin(40 degrees) ≈ 5.13 ft
B = (8 ft + 2 ft) * sin(60 degrees) ≈ 9.80 ft
Oh dear! It looks like B is greater than A, indicating that the beam won't quite clear those wires on its way to standing up straight. We might need to come up with a different plan to avoid some shocking encounters.
Remember, safety first, and always be careful when dealing with heights, wires, and construction sites!
To determine whether the beam will clear the wires when it stands up straight, we need to calculate the height of the beam when it is fully standing and compare it to the height of the wires.
Let's use trigonometry to solve this problem.
We are given that when the beam makes an angle of 40 degrees with the ground, the top of the beam is 8 ft above the ground. Let's call this distance "x".
Using trigonometry, we can express the height of the beam as:
x = height / sin(40)
Now, we need to calculate the height of the beam when it is fully standing. Since we want to find the height of the beam when it makes an angle of 90 degrees with the ground, we can use a right triangle.
Let's call the height of the beam when fully standing "h" and the distance from the top of the beam to the base "b".
In the right triangle formed by the beam when fully standing, the hypotenuse is the height of the beam (h), the base is the distance from the top of the beam to the ground (x), and the perpendicular side is the distance from the top of the beam to the wires (2 ft).
Using Pythagorean theorem, we can express this relationship as:
h^2 = x^2 + b^2
Since we know that when the beam is at an angle of 40 degrees, x = 8 ft, we can substitute this value in the equation:
h^2 = (8 ft)^2 + b^2
Now, we need to find the value of b in order to solve for h.
When the beam makes an angle of 60 degrees with the ground, we know that the wires are 2 ft above the top of the beam. This means that b = x + 2 ft.
Substituting this value into the equation, we have:
h^2 = (8 ft)^2 + (x + 2 ft)^2
Now, let's calculate the values and determine whether the beam will clear the wires.
x = height / sin(40) = 8 ft / sin(40) = 12.21 ft (approx.)
Substituting this value into the equation:
h^2 = (8 ft)^2 + (12.21 ft + 2 ft)^2
= 64 ft^2 + 196.84 ft^2
= 260.84 ft^2
Now, let's calculate the value of h:
h = √260.84 ft^2 = 16.13 ft (approx.)
Therefore, the height of the fully standing beam is approximately 16.13 feet.
Since the height of the wires is 2 ft above the top of the beam, and the height of the fully standing beam is 16.13 ft, we can conclude that the beam will clear the wires when it stands up straight.
To determine if the beam will clear the wires, we need to find the height of the beam when it reaches an angle of 60 degrees.
Let's break down the problem and solve it step by step:
Step 1: Determine the initial height of the beam when it starts to be lifted. From the problem, we know that when the beam makes an angle of 40 degrees with the ground, the top of the beam is 8 ft above the ground. This gives us the initial height of the beam.
Step 2: Calculate the height of the beam when it reaches an angle of 60 degrees. We can create a right triangle with the height of the beam at 40 degrees, the height of the beam at 60 degrees (h), and the difference in angles between 60 and 40 degrees (20 degrees).
Step 3: Use trigonometry (specifically, the tangent function) to calculate the height at 60 degrees. Since we have the opposite (h) and adjacent (8 ft) sides, we can use the tangent function to find the angle: tan(20 degrees) = h / 8 ft.
Step 4: Solve for h by rearranging the equation: h = 8 ft * tan(20 degrees).
Step 5: Substitute the value of h into the equation and calculate the final height of the beam at 60 degrees.
Step 6: Compare the final height of the beam at 60 degrees with the height of the wires (2 ft). If the height of the beam is greater than the height of the wires, the beam will clear the wires. If it is less, the beam will hit the wires.
Now let's calculate the final height of the beam:
Step 3: h = 8 ft * tan(20 degrees)
h = 8 ft * 0.36397 (using a calculator)
h ≈ 2.9118 ft
The final height of the beam at 60 degrees is approximately 2.9118 ft.
Step 6: Since the final height of the beam at 60 degrees (2.9118 ft) is greater than the height of the wires (2 ft), the beam will clear the wires on its way to standing up straight.
Therefore, the beam will not hit the wires.