Problem 20 - Intermediate Algebra

Question

Machining. Each pass through a lumber plane shaves off 0.015 inch of thickness from a board. How many times must a board, originally 0.875 inch thick, be run through the planer if a board of thickness 0.74 inch is desired?

Step-by-Step Solution

Answer

The board must be run through the planer 9 times.

Step 1: Determine Initial Thickness

The initial thickness of the board is 0.875 inches.

Step 2: Determine Desired Final Thickness

The desired thickness of the board after machining is 0.74 inches.

Step 3: Calculate Thickness Reduction Required

Find the total reduction in thickness required by subtracting the final thickness from the initial thickness: \[0.875 - 0.74 = 0.135\text{ inches}\]

Step 4: Determine Thickness Removed Per Pass

Each pass through the planer removes 0.015 inches of thickness from the board.

Step 5: Calculate Number of Passes Required

Divide the total thickness reduction by the thickness removed per pass to find the number of passes:\[\frac{0.135}{0.015} = 9\] This indicates 9 passes are required.

Key Concepts

Understanding the Lumber Plane Process

When it comes to woodworking, the lumber plane process is essential for achieving precise thickness in boards. A lumber plane, often referred to as a planer, is a woodworking machine that shaves off the surface of the wood to reduce its thickness. This makes the surface smooth and even, which is crucial for ensuring the wood fits perfectly in its final application.
There are several stages in this process:

• The board is fed through the planer, where rotating blades shave off small amounts of wood thickness with each pass.
• The machine's blades are set to remove a specific thickness per pass, ensuring consistency.
• After multiple passes, the board reaches the desired thickness, with a smooth and even surface.

This process is commonly used when boards need to match specific thicknesses, especially in cabinetry and furniture making.

Performing Thickness Reduction Calculations

Calculating the amount of thickness to reduce from a piece of lumber involves determining both the initial and desired thickness.
Begin with the initial thickness, which is how thick the board is before processing.
Next, identify the desired final thickness — how thick you want the board to be after being planed.
The reduction in thickness required is calculated by subtracting the desired thickness from the initial thickness:

• Example: Given an initial thickness of 0.875 inches and a desired thickness of 0.74 inches, the required thickness reduction is 0.875 - 0.74 = 0.135 inches.

This value represents the total amount of wood that needs to be shaved off the board to achieve the desired thickness.

Mastering Division in Word Problems

Division is crucial in solving word problems related to machining and other applications. In machining, division helps determine how many times an operation needs to be repeated to achieve a specific goal.
For example, to find how many passes a board must go through a planer, you divide the total thickness reduction needed by the thickness removed per pass:

• If 0.135 inches need to be shaved off and each pass removes 0.015 inches, the calculation is \( \frac{0.135}{0.015} = 9 \).

This result means that nine passes are required.
Understanding how to set up and solve these division problems is important for precise and accurate machining. Being able to interpret the numbers and steps in real-life contexts enhances your ability to tackle similar challenges efficiently.