Final answer:
To find the minimum length of the first piece of the pipe, set up the inequality x + 2x + (2x + 1) ≥ 21, solve for x, and find that x ≥ 4 feet is the minimum length for the first piece.
Step-by-step explanation:
The student needs to determine the minimum length of the first piece of a pipe that is at least 21 feet long when it is cut into three pieces with the given conditions.
Let's denote the length of the first piece as x feet. According to the problem, the second piece must be twice as long as the first piece, so it will be 2x feet. The third piece must be one foot longer than the second piece, which makes it 2x + 1 feet. The total length of the pipe which is at least 21 feet can be expressed by the following inequality:
x + 2x + (2x + 1) ≥ 21
Combining like terms, we get:
5x + 1 ≥ 21
Subtracting 1 from both sides of the inequality gives us:
5x ≥ 20
Dividing both sides by 5 gives us:
x ≥ 4
Therefore, the minimum length of the first piece is 4 feet.