Problem 11 - Technical Mathematics with Calculus

Question

A 1.00 -in. cube of steel is placed in a surface grinding machine, and the vertical feed is set so that 0.0050 in. of metal is removed from the top of the cube at each cut. How many cuts are needed to reduce the volume of the cube by 0.50 in. \(^{3} ?\)

Step-by-Step Solution

Answer

100 cuts are needed to reduce the volume of the cube by 0.50 in.(3).

Step 1: Understanding the problem

We need to determine how many cuts it will take to reduce the volume of a 1.00-inch steel cube by 0.50 cubic inches. Each cut removes 0.0050 inches from the top of the cube.

Step 2: Calculating the area removed per cut

To find out the volume removed per cut, we calculate the area of the top face of the cube that is removed by each cut. Since the cube is 1.00 inch on each side, the area of the top face before any cuts is 1.00 inch squared. Multiplying this by the depth of the cut (0.0050 inches) gives us the volume removed per cut: \(1.00 \text{ in}^2 \times 0.0050 \text{ in} = 0.0050 \text{ in}^3\) per cut.

Step 3: Calculating the number of cuts needed

To achieve the desired volume reduction of 0.50 cubic inches, we divide this amount by the volume removed per cut. \(0.50 \text{ in}^3 \div 0.0050 \text{ in}^3 = 100\) cuts are needed to reduce the volume by 0.50 cubic inches.

Key Concepts

Volume Calculation

Volume calculation is a fundamental aspect of geometry that helps us determine the three-dimensional space occupied by an object. For solid shapes like cubes, the volume is found by multiplying the length by the width and the height. In our case, we have a steel cube with each side measuring 1.00 inch.

Calculating the volume reduction, therefore, involves understanding how much space is being taken away with each surface grinding cut. Since 0.0050 inch is removed downwards, per cut, it's crucial to grasp that this actually removes a 0.0050 inch thick layer of the entire top face of the cube, which is 1.00 inch squared. The removed volume per cut is simply the product of the area of the top face and the depth removed, giving us \(0.0050 \text{ in}^3\).

Remember, every cut will always remove the same amount of material, as the cross-sectional area of the top face remains 1.00 inch squared, despite the overall volume decreasing. Continuing to apply these fundamentals helps in visualizing and mastering the calculation of complicated volumes for various shapes and materials in technical mathematics.

Surface Grinding

Surface grinding is a common machining process that involves removing material from the surface of a workpiece using a rotating abrasive wheel. In our exercise, a cube of steel undergoes this process. Surface grinding is particularly useful for achieving a high-quality finish and bringing materials to a very specific size or surface area.

To appreciate the precision of surface grinding in industrial applications, consider that each pass of the grinder over the material removes a thin layer—markedly just 0.0050 inch in our problem. This highlights two critical points: the significance of consistent material removal for achieving accurate dimensions and the considerable attention to detail required for such mathematical calculations in technical settings.

Understanding this process is not only vital for solving mathematical problems but also for comprehending manufacturing techniques that require precise control over material dimensions and surface conditions.

Mathematical Problem Solving

Mathematical problem solving is an intellectual process that involves comprehending given information, formulating a strategy, and carrying out a series of logical steps to arrive at a solution. In technical mathematics, each problem demands a clear understanding and application of relevant mathematical principles and concepts.

In our exercise, we took a systematic approach to problem-solving: we began by comprehending the problem's goal, proceeded to calculate the volume removed per cut, and finally determined the number of cuts needed to reach the desired volume reduction. This kind of logical reasoning is a cornerstone of mathematics. Encouraging a step-by-step method ensures that complex operations are broken down into manageable tasks. This practice enhances cognitive processes and promotes accuracy in problem-solving, which is especially crucial in fields where precision is paramount, such as engineering and manufacturing.

Successful problem-solving also involves checking the reasonableness of an answer and having awareness of the practical implications of the solution, such as the feasibility of the number of cuts in a real-world manufacturing process.