Hi Student3! Thank you for the opportunity to help you with your question!

We first need to set up variables for each piece of rope. Let the 1st piece be x, the 2nd be y and the third be z.

So the equation of the rope is x + y + z = 240.

We are told that the 1st piece is twice as long as the 2nd piece, so x=2y OR y=0.5x since this also means the 2nd piece is half as long as the 1st piece.

We are also told that the 3rd piece is twice as long as the 1st piece, so z=2x.

So here's what we have so far:

x = x

y=0.5x

z=2x

We can now substitute these values into the equation of the rope and solve for x.

x + y + z = 240

x + 0.5x + 2x = 240

3.5x = 240 (Combine like terms)

x = 240/3.5 (Divide both sides by 3.5 to isolate the variable)

x = 68.6 (Rounded to 1 decimal place)

Now, since the 1st piece is about 68.6 feet, we can figure out the lengths of the other two pieces.

The 2nd piece is half the length of the 1st piece so the 2nd piece is about 34.3 feet.

The 3rd piece is twice the length of the 1st piece so the 3rd piece is about 137.1 feet.

Hence, the longest piece is about 137.1 feet.

Note: we can check our answer by adding the three values together, 68.6 + 34.3 + 137.1 = 240 feet! :-)

Please let me know if you need any clarification. I'm always happy to answer your questions!