Problem 65 - 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. Copper: \(H=140\)

Step-by-Step Solution

Answer

The depth of the indentation for copper is 0.12mm.

Step 1: Analysis of the problem

The hardness of Copper \(H\) is given as 140. We have to plug this value into the formula \(Hd^{2}=1.89\) and solve for \(d\).

Step 2: Solving for d

First rewrite the formula given: \(140d^{2}=1.89\). Now solve for \(d\), by dividing both sides of the equation by 140. This gives: \(d^{2} = 1.89/140\). In order to isolate \(d\), take the square root of both sides of the equation: \(d = \sqrt{1.89/140}\).

Step 3: Calculation and Rounding

By using a calculator to solve the above, we find that \(d = 0.1153452848805752\). In order to round this number to the nearest hundredth, reduce it to two decimal places, resulting in \(d = 0.12\) mm. This is the depth of the indentation for Copper.

Key Concepts

Mineral Hardness

Mineral hardness is an essential property in geology and materials science. It helps us understand how resistant a mineral is to deformation or scratching. The concept was first formalized through the Mohs scale of mineral hardness, which ranks minerals based on their ability to scratch another mineral. Mohs scale uses a soft mineral like talc as the lowest value, and a hard mineral like diamond as the highest.
However, the Vickers scale introduces a more scientific and precise method of measuring mineral hardness. It involves the use of a diamond indenter, shaped like a pyramid, which makes an impression on the mineral surface under a specific force.

• The smaller the indentation, the harder the mineral.
• Hardness values are crucial for determining the utility and durability of minerals in various industries.

This precision makes the Vickers scale highly valuable in research, as it provides a quantitative measurement rather than just a qualitative one.

Vickers Scale

The Vickers scale is an innovative way to measure the hardness of materials, including minerals. Named after the engineer George E. Vickers, this method is known for its accuracy. To measure hardness, a diamond with a square-based pyramidal shape is pressed into the material's surface at a defined force.

• The depth of the indentation is inversely related to the material's hardness.
• This depth is then used in a calculation to provide a Vickers Hardness Number, or HV.

The equation involving the Vickers scale often relates hardness to the depth of the indentation, as seen in the exercise where the formula is given as \(H d^2 = 1.89\). By arranging the equation properly, you can solve for any of the variables, providing important insights about the material's properties.
One of the benefits of the Vickers scale is that it is suitable for a wide range of materials and can provide consistent and comparable results across different tests.

Square Roots

Square roots are a fundamental concept in mathematics and are used frequently in algebraic equations, like the one found in the example exercise. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4, because \(4^2 = 16\).
In the context of the formula \(140d^2 = 1.89\), finding the depth \(d\) involves calculating the square root of both sides after dividing by 140. By isolating \(d^2\), we take the square root to solve for \(d\).

• These calculations are often simplified using calculators for precision.
• Understanding square roots helps in solving many types of algebraic equations as they appear in various fields.

Mastering square root calculations can greatly aid in comprehending and solving complex mathematical problems, enabling students to handle even the most challenging equations with ease.