A carpenter cuts a 12-foot piece of lumber into four pieces using a chop saw. The first piece is $37 \frac{3}{4}$ inches long. The second piece is $25 \frac{3}{8}$ inches long. The third piece is 52 inches long. Each saw cut wastes $\frac{1}{8}$ inch of wood as it cuts and creates sawdust. How long is the remaining piece?
Answer (2)
Convert the total length of the lumber to inches: 12 × 12 = 144 inches.
Calculate the total waste from the three cuts: 3 × 8 1 = 0.375 inches.
Calculate the sum of the lengths of the first three pieces: 37.75 + 25.375 + 52 = 115.125 inches.
Subtract the total waste and the sum of the lengths from the total length: 144 − 0.375 − 115.125 = 28.5 inches. The length of the remaining piece is 28.5 inches.
Explanation
Problem Overview We are given a 12-foot piece of lumber that is cut into four pieces. We know the lengths of the first three pieces and the amount of wood wasted per cut. Our goal is to find the length of the remaining piece.
Convert to Inches First, we need to convert the total length of the lumber to inches since the lengths of the pieces are given in inches. There are 12 inches in a foot, so the total length is: 12 feet × 12 foot inches = 144 inches
Calculate Total Waste Next, we need to calculate the total amount of wood wasted during the cuts. There are three cuts, and each cut wastes 8 1 inch of wood. So the total waste is: 3 cuts × 8 1 cut inch = 8 3 inches = 0.375 inches
Sum of Known Lengths Now, we need to find the sum of the lengths of the first three pieces: 37 4 3 + 25 8 3 + 52 = 37.75 + 25.375 + 52 = 115.125 inches
Calculate Remaining Length Finally, we subtract the total waste and the sum of the lengths of the three pieces from the total length of the lumber to find the length of the remaining piece: 144 − 0.375 − 115.125 = 28.5 inches
Final Answer Therefore, the length of the remaining piece is 28.5 inches.
Examples
Carpenters frequently need to calculate how to efficiently cut materials to minimize waste. This problem demonstrates how to calculate the length of a remaining piece of lumber after several cuts, accounting for the material lost during each cut. This is crucial for projects like building furniture, framing walls, or any task where precise material usage is essential. By understanding these calculations, carpenters can optimize their material usage, reduce costs, and ensure accurate project completion.
The length of the remaining piece of lumber after three cuts and accounting for sawdust is 28.5 inches. This was calculated by converting the total length to inches, determining the waste from cuts, and subtracting the total used length from the original length. In total, the remaining piece measures 28.5 inches in length.
;
