Problem 63 - 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. Galena: \(H=80\)

Step-by-Step Solution

Answer

The depth of indentation for Galena on the Vickers scale is approximately \(0.15\) mm.

Step 1: Rearrange the equation

To find the indentation depth \(d\) for any given hardness \(H\), you need to rearrange the equation \(Hd^{2}=1.89\) to make \(d\) the subject of the formula. This would become \(d=\sqrt{\frac{1.89}{H}}\).

Step 2: Substitute Hardness Value

In this scenario for the mineral Galena, the hardness \(H\) is 80. Substituting this value into the formula, it will become \(d=\sqrt{\frac{1.89}{80}}\).

Step 3: Calculate and Round Up

Using a calculator to compute the value would end up with \(d \approx 0.153\), but you're asked to round up to the nearest hundredth of a millimeter, leading to the final value of \(d \approx 0.15\) mm.

Key Concepts

Vickers scale

When it comes to assessing mineral hardness, the Vickers scale is a widely used method. This system is based on the principle of indentation hardness measurement, which involves using a diamond indenter with a pyramid shape. The idea is to apply force to the mineral surface and record how it resists deformation.

• The Vickers scale provides a numerical value that reflects the hardness of the mineral.
• The smaller the indentation, the harder the mineral is considered to be.
• Hardness values on the Vickers scale are compared to assess different materials.
• This method is preferred because it can be used on a variety of materials, including metals and ceramics, as well as minerals.

Understanding the Vickers scale is essential when studying different materials to predict their durability and resistance to wear and scratching.

mineral hardness

Mineral hardness refers to the resistance a mineral exhibits against being scratched or indented. In the context of the Vickers scale, hardness is quantitatively measured. Here's how it generally works:

• The diamond indenter is pressed into the mineral's surface.
• Depending on how easily or hard the mineral allows the indenter to make an impression, its hardness is determined.
• Hard minerals require more force to leave an indentation than softer ones.
• This characteristic is critical in understanding a mineral's applications in various fields such as jewelry or construction.

For instance, diamonds are renowned for their extreme hardness, making them ideal in cutting tools and wear-resistant applications.

indentation depth

Indentation depth is a crucial measurement in determining mineral hardness **using the Vickers scale**. It relates directly to how hard or soft a material is under a given load. With minerals, like Galena, measuring indentation depth helps us understand their nature.

• Indentation depth is inversely proportional to mineral hardness; smaller indentations indicate harder minerals.
• The depth is typically measured in millimeters and is calculated based on the load applied during the test.
• This calculation involves knowing the applied load and the specific geometry of the indenter, which influences how deep the indentation will be under a given hardness level.

Accurate measurement of indentation depth is essential for ensuring the reliability and validity of hardness tests, allowing for meaningful comparisons across different minerals.

square root calculation

Square root calculation is an essential mathematical skill, especially in problems involving geometrical relationships and physical measurements, like the Vickers hardness test.

In these tests, the formula used to relate hardness and indentation depth is often in the form of a square relationship like \( H d^{2} = 1.89 \).

• To isolate the variable of interest, such as indentation depth \(d\), involves taking the square root.
• This is done by rearranging the equation to express \(d\) in terms of its square root: \( d = \sqrt{ \frac{1.89}{H} }\).
• Carrying out this calculation provides the depth of indentation for known hardness values, vital for accurate assessments of material properties.
• Using a calculator for square root calculations is common practice to ensure precision, particularly when dealing with scientific data.

Mastery of square root calculations is not only useful in scientific tasks but also enhances mathematical reasoning skills.