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?
Can I please get help with how to specifically solve this? I am very confused and I need step by step guidance.

As the problem is stated, of course they will because as it rises the horizontal distance from the pivot point at the ground decreases. If the wires are exactly above the tip of the bean at 60 deg, then they will be closer to the pivot point at more than 60deg

Therefore I assume you have left out the horizontal overlap at 60 degrees and you need to figure out haw far back the beam moves as it pivots up those last two feet

at 8 feet up
sin 40 = 8/length of beam
so L = 12.45 ft length

then at 60 degrees
sin 60 = h/12.45
h = 10.78 high
and horizontal distance is
12.45 cos 60 = 6.23

now at h = 2+10.78 = 12.78
BUT that is higher than the beam is long :)
No way it will hit the wires.

I think the answer is 12.78. But I'm not 100% honest.

To solve this problem, we can use trigonometry (specifically the tangent function). Here's a step-by-step guide on how to solve it:

Step 1: Draw a diagram of the situation described in the problem. Label the given measurements - 8 ft for the height of the beam when it's at a 40-degree angle, and 2 ft for the height of the wires above the beam when it's at a 60-degree angle.

Step 2: Break down the problem into two separate triangles. The first triangle is formed by the beam and the ground, with an angle of 40 degrees and a height of 8 ft. The second triangle is formed by the beam, the ground, and the wires, with an angle of 60 degrees and a height of 2 ft.

Step 3: Using the tangent function, calculate the length of the base of the first triangle (the distance from the beam's base to the point where the rope is tied). The tangent of an angle is equal to the opposite side (8 ft) divided by the adjacent side (length of the base). In this case, we can solve for the base length using the formula: tan(40 degrees) = 8 ft / base length.

Step 4: Solve for the base length by rearranging the equation from Step 3: base length = 8 ft / tan(40 degrees). Use a calculator to find the approximate value of the tangent of 40 degrees, and then divide 8 ft by that value to get the length of the base.

Step 5: Calculate the height of the beam when it's standing up straight by subtracting the length of the base (from Step 4) from the total height of the beam. This will give you the maximum height the beam will reach when it's standing up straight.

Step 6: Compare the calculated height from Step 5 with the height of the wires (2 ft). If the calculated height is greater than the height of the wires, then the beam will clear the wires. If the calculated height is equal to or less than the height of the wires, then the beam will hit the wires.

That's it! By following these steps, you will be able to determine whether the beam will clear the wires on its way to standing up straight.

To solve this problem, we can use trigonometry and create a diagram to represent the situation. Let's follow these steps to find the answer:

Step 1: Draw a diagram:
Draw a vertical line to represent the ground. Label the bottom point as the base of the beam and the top point as the top of the beam. Draw a horizontal line passing through the top of the beam to represent the level of the wires. Label this line as the wires.

Step 2: Determine the necessary measurements:
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. Additionally, when the beam makes an angle of 60 degrees with the ground, the wires are 2 ft above the top of the beam.

Step 3: Calculate the height of the beam at 40 degrees:
Using trigonometry, we can use the given information to find the height of the beam at a 40-degree angle. We can use the sine function:

sin(40°) = opposite/hypotenuse

In this case, the opposite side is the height of the beam and the hypotenuse is the distance from the top of the beam to the base. Let's label the distance from the top of the beam to the base as "d1." We can set up the equation:

sin(40°) = 8/d1

Now, solve for d1 by rearranging the equation:

d1 = 8/sin(40°)

Use a calculator to find the value of d1.

Step 4: Find the height of the wires:
We are given that when the beam makes an angle of 60 degrees with the ground, the wires are 2 ft above the top of the beam. To find the height of the wires, we need to subtract the height of the beam from this information. Let's label the height of the wires as "h_wires."

h_wires = 2 - 8

Step 5: Calculate the distance from the top of the beam to the base at a 60-degree angle:
Using trigonometry again, we can use the height of the wires to find the distance from the top of the beam to the base when the beam makes a 60-degree angle. We can set up the equation:

sin(60°) = h_wires/d2

Now, solve for d2 by rearranging the equation:

d2 = h_wires/sin(60°)

Use a calculator to find the value of d2.

Step 6: Compare the distances d1 and d2:
Now that we have obtained the distances d1 and d2, we can compare them. If d1 is greater than d2, it means the beam will not clear the wires on its way to standing up straight. If d1 is less than or equal to d2, it means the beam will clear the wires.

I hope this step-by-step guide helps you to solve the problem. Let me know if you have any further questions!