Problem 64 - Algebra 1: Concepts and Skills

Question

Use the following information. Mineralogists use the Vickers scale to measure the hardness of minerals. The hardness \(H\) of a mineral can be determined by hitting the mineral with a pyramid-shaped diamond and measuring the depth \(d\) of the indentation. The harder the mineral, the smaller the depth of the indentation. A model that relates mineral hardness with the indentation depth (in millimeters) is \(H d^{2}=1.89\). Use a calculator to find the depth of the indentation for the mineral with the given value of \(H .\) Round to the nearest hundredth of a millimeter. Platinum: \(H=125\)

Step-by-Step Solution

Answer

The depth of indentation for Platinum with a hardness of 125 on the Vickers Scale is approximately 0.12 millimeters.

Step 1: Write Down The Given Formula and The Given Values

We have the formula \(H d^{2}=1.89\). The given hardness \(H\) for Platinum is 125.

Step 2: Rearrange The Formula to Solve for Depth \(d\)

In order to solve for \(d\), we first need to divide both sides of the equation by \(H\). So, we have \(d^{2}= \frac{1.89}{H}\).

Step 3: Substitute The Given Hardness \(H\) into The Equation

We can substitute \(H=125\) into the equation: \(d^{2}= \frac{1.89}{125}\).

Step 4: Use a Calculator to Simplify The Expression on The Right Side of The Equation

When we do the division, we find that \(d^{2} \approx 0.01512\).

Step 5: Solve for The Depth \(d\)

In the equation \(d^{2} \approx 0.01512\), we can solve for \(d\) by taking the square root of both sides of the equation resulting in \(d \approx \sqrt{0.01512}\).

Step 6: Use a Calculator to Find The Numerical Value of \(d\) to The Nearest Hundredth of a Millimeter

Use a calculator to find the value of \(\sqrt{0.01512}\) and round to the nearest hundredth of a millimeter. The result will be approximately \(d \approx 0.12\) millimeters.

Key Concepts

Hardness Measurement

Hardness measurement is a method used to determine how resistant a material is to deformation or penetration. It's a crucial concept in materials science, especially in the context of minerals and metals. When we say a material is hard, we mean that it's tough to scratch or dent. In real-world applications:

• Hardness can affect the durability of products.
• It is essential for choosing materials for construction and manufacturing.

Methods such as the Rockwell, Brinell, and Vickers hardness tests are commonly used to assess hardness. These tests involve pressing an indenter into the surface of the material and measuring the size of the mark left. The smaller the mark, the harder the material. Since different materials require different testing methods, understanding hardness measurement is vital in ensuring the right equipment and techniques are used.

Vickers Scale

The Vickers scale is a popular method for measuring the hardness of materials, particularly metals and minerals. Unlike other hardness tests, the Vickers test uses a diamond-shaped (pyramid-like) indenter. This ensures consistent results across a wide range of materials.

• The test is versatile and can be used on nearly any type of solid material.
• It provides a precise measure of hardness, as it can be applied to thin materials and small areas.

To measure hardness using the Vickers scale, the formula used is: \[ H d^{2} = 1.89 \]Where:

• \(H\) is the hardness of the material.
• \(d\) is the depth of the indentation.

The result of the test is a Vickers hardness number (HV), which can provide insights into the material's properties, such as wear resistance and ductility. The Vickers test plays an essential role in research and industry due to its adaptability to different conditions and materials.

Indentation Depth

Indentation depth refers to the distance or depth that an indenter penetrates into a material during a hardness test. It's a key part of assessing a material's hardness on the Vickers scale. Here's why it's critical:

• A shallower indentation depth indicates a harder material.
• Understanding indentation depth helps predict how a material will perform under stress.

The process of finding the indentation depth in this context involves rearranging the equation: \[ d^{2} = \frac{1.89}{H} \]Using this formula, by substituting the value of \(H\), you get \(d^{2}\). Taking the square root will yield \(d\), which in this exercise comes out to 0.12 mm when rounded to the nearest hundredth. This number represents how deep the pyramid-shaped indenter sank into the material surface. The concept of indentation depth not only provides the hardness level but also relates to the physical toughness of the material being tested.