a 210 ft. long rope is cut into 3 pieces. the first piece of rope is 3 times as long as the second rope. the 3rd piece of rope is 2 times as long as the second piece of rope. What is the lenght of the longest piece of rope?

Let x = 2nd Rope, then 3x = first and 2x = third.

x + 3x + 2x = 210

Solve for x.

35

A 180 ft long board is cut into 3 pieces. The second piece of board is twice as long as the first. The third piece of board is six times as long as the first.

What is the length of the first piece of the board?

105ft.

To find the length of the longest piece of rope, we need to determine the lengths of all three pieces first.

Let's represent the length of the second piece as x.

According to the given information, the first piece is three times as long as the second piece, so its length is 3x.

Similarly, the third piece is two times as long as the second piece, so its length is 2x.

Now, we know the total length of all three pieces is 210 ft. We can set up an equation to represent this:

x + 3x + 2x = 210

Simplifying the equation, we combine the like terms:

6x = 210

To solve for x, divide both sides of the equation by 6:

x = 210 / 6 = 35

Now, we know the length of the second piece is 35 ft.

To find the lengths of the first and third pieces, we can substitute this value back into the equations we derived earlier:

The length of the first piece = 3x = 3 * 35 = 105 ft.
The length of the third piece = 2x = 2 * 35 = 70 ft.

Finally, to find the length of the longest piece, we compare the lengths of all three pieces:

Longest piece = max(35, 105, 70) = 105 ft.

Therefore, the length of the longest piece of rope is 105 ft.