Certainly! Let's work through this problem step by step.
### Part (a)
1. Identifying the positions of the ends of the board:
The carpenter places the measuring tape such that one end of the board is at 24 inches and the other end is at 8 inches on the tape.
2. Calculating the length of the board:
To find the length of the board, subtract the position of the smaller measurement (the 8-inch end) from the position of the larger measurement (the 24-inch end):
[tex]\[
\text{Length of the board} = 24 \, \text{inches} - 8 \, \text{inches} = 16 \, \text{inches}
\][/tex]
So, the length of the board is 16 inches.
### Part (b)
3. Finding the midpoint of the board:
To cut the board into two pieces of equal length, we need to find the midpoint of the board. The midpoint can be found by measuring half the length of the board.
[tex]\[
\text{Midpoint} = \frac{16 \, \text{inches}}{2} = 8 \, \text{inches}
\][/tex]
4. Determining the position on the measuring tape:
Since the measuring tape is still in the same position as described in part (a), with one end at 24 inches and the other at 8 inches, we need to find the position on the measuring tape where this midpoint falls. We add the 8-inch starting point of the board to the midpoint we calculated:
[tex]\[
\text{Marking point on tape} = 8 \, \text{inches} + 8 \, \text{inches} = 16 \, \text{inches}
\][/tex]
So, the carpenter should mark the board at the 16-inch point on the tape to make the cut for two equal pieces.
### Summary
- Length of the board: 16 inches
- Marking point for the cut: 16 inches
These are the detailed answers for parts (a) and (b) of the question.
