Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

British sterling silver is a copper-silver alloy that is \(7.5 \%\) copper by weight. How many grams of pure copper and how many grams of British sterling silver should be used to prepare 200 grams of a copper-silver alloy that is \(10 \%\) copper by weight?

Step-by-Step Solution

Answer

Use 194.59 grams of sterling silver and 5.41 grams of pure copper.

Step 1: Define Unknowns

Let \( x \) be the grams of British sterling silver, and \( y \) be the grams of pure copper needed. Since we want to prepare 200 grams of alloy, we have the equation \( x + y = 200 \).

Step 2: Set Up Copper Content Equation

British sterling silver contains \( 7.5\% \) copper, so the copper content in sterling silver is \( 0.075x \). Pure copper is \( y \) grams. The desired alloy is \( 10\% \) copper, which equals \( 0.10 \times 200 = 20 \) grams. Therefore, we have the equation \( 0.075x + y = 20 \).

Step 3: Solve the System of Equations

We have two equations: \( x + y = 200 \) and \( 0.075x + y = 20 \). Subtract the second equation from the first to get: \( x + y - (0.075x + y) = 200 - 20 \). This simplifies to \( 0.925x = 180 \). Solving for \( x \), we get \( x = \frac{180}{0.925} \approx 194.59 \).

Step 4: Calculate Pure Copper

Substitute \( x \approx 194.59 \) back into the equation \( x + y = 200 \) to find \( y \). Thus, \( y = 200 - 194.59 = 5.41 \).

Step 5: Verification

Verify that the values satisfy both conditions. For copper content: \( 0.075 \times 194.59 + 5.41 \approx 14.59 + 5.41 = 20 \), which is correct. The weight condition is also satisfied since \( x + y = 200 \).

Key Concepts

System of Equations

When solving real-world problems, especially in algebra, a system of equations can be incredibly useful. This method involves setting up two or more equations that share common variables, and solving them simultaneously.

• A system of equations helps us find the values of multiple variables at once.
• It consists of two or more equations that share the same set of unknowns.
• To solve this, you can use methods like substitution, elimination, or graphing.

In the exercise, we used two equations to solve for the unknown amounts of British sterling silver and pure copper needed to create a desired alloy. The equations reflect the total alloy weight and its copper content. By solving these equations together, we can determine the proportions of each component necessary to reach the desired result.

Copper-Silver Alloy

A copper-silver alloy is a mixture of copper and silver that combines the properties of both metals. Alloys are often used in jewelry and various industrial applications because they offer enhanced characteristics, such as strength or resistance to tarnishing.

• British sterling silver is a popular copper-silver alloy consisting of 92.5% silver and 7.5% copper by weight.
• This composition offers durability and a brilliant shine, making it ideal for crafting fine silverware and jewelry.

Understanding the composition of an alloy, like British sterling, is crucial when determining how to achieve a specific mixture for new applications. The problem in the exercise aims to create a specific copper-silver alloy composition by adjusting the amounts of its components.

Percentage Concentration

Percentage concentration refers to the amount of a particular substance within a mixture, expressed as a percentage of the total weight or volume. It is vital in chemistry and manufacturing processes for specifying the exact composition of a mixture.

• The percentage concentration gives an easy understanding of the proportion of components in a mixture.
• In the exercise, the desired copper concentration in the alloy is 10% by weight.
• By knowing the percentage concentration, we can accurately measure and mix materials to achieve the desired component ratio.

In the problem, understanding the copper concentration in each component (pure copper and British sterling silver) allows us to use algebra to find the correct quantities needed to meet the desired 10% copper concentration.

Chemical Mixtures

Chemical mixtures involve combining two or more substances in such a way that they maintain their individual chemical properties. These mixtures, such as alloys, are common in various scientific and industrial fields.

• Mixtures can be homogeneous, where the composition is uniform, or heterogeneous, where the composition varies throughout.
• An alloy like the copper-silver one in our exercise is a homogeneous mixture – it has a consistent composition throughout.

The exercise requires creating a new alloy mixture with a specific copper-silver ratio, demonstrating practical applications of chemistry in product formulation and manipulation. By understanding chemical mixtures, we can predict how the components will interact and adjust processes to ensure a quality outcome.