Question

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain bronze alloy containing \(4 \%\) tin is to be added to 351 lb of bronze containing \(18 \%\) tin to produce a new bronze containing \(15 \%\) tin. How many pounds of the \(4 \%\) bronze are required?

Short answer

Approximately 95.7 pounds of the 4% bronze are required.

Step-by-step solution

Define Variables

Let the amount of the 4% tin bronze to be added be represented by x (in pounds).

Set Up the Equation

The total amount of tin from both alloys must equal the amount of tin in the final alloy. This can be set up as an equation: 0.04x (tin from the 4% bronze) + 0.18(351 lb) (tin from the 18% bronze) = 0.15(x + 351 lb) (tin in the final bronze).

Solve the Equation

Distribute the percentages and combine like terms to solve for x: 0.04x + 63.18 = 0.15x + 52.6563.18 - 52.65 = 0.15x - 0.04x10.53 = 0.11xx = 95.7 lb.

Verify the Solution

Check that the resulting mix has 15% tin by substituting x back into the original equation: 0.04(95.7) + 0.18(351) 3.828 + 63.18 = 0.15(95.7 + 351) 67.008 0.15(446.7) = 67.005 The values are within the significant digits range, verifying the solution is correct.

Key concepts

Algebraic Equations

When solving percentage problems in algebra, we often use algebraic equations to find unknown values. An algebraic equation is a mathematical statement that shows the equality of two expressions. They are formed by combining variables, numbers, and operations such as addition and multiplication. In our case, the equation helps to calculate the quantity of a substance in a mixture.

For the bronze alloy problem, the equation was set up based on the condition that the total amount of tin in the final mixture must equal the total amount of tin from the original alloys. Using the variable 'x' to represent the unknown amount of the 4% tin bronze needed, the equation was expressed as a balance between the tin from the mixed bronzes and the tin in the resulting alloy. This is a classic example of how algebra enables us to solve for an unknown in a real-world context, such as determining quantities in a compound or mixture.

Significant Digits

Significant digits (or significant figures) are a critical concept when dealing with precision in measurements and computations in algebra. These are the digits in a number that carry meaning contributing to its precision. This includes all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion of a number.

In the context of the bronze alloy problem, the problem statement instructs to work to three significant digits. This means that our final answer needs to be precise to three important figures. For instance, when we calculated the weight of the 4% tin bronze alloy to be 95.7 lb, we rounded our answer to three significant figures, because the problem demands this level of precision. Understanding significant digits is important to ensure that any final answer in a calculation is as accurate as the precision of the numbers provided.

Weight Computations

Weight computations involve calculating the mass of an object or a substance and are common in science and mathematical problems. In our original exercise, weight computation is used to determine the amount in pounds of a bronze alloy that is needed to achieve a desired composition percentage. The computation takes into consideration the weight of the existing alloy and the percentage composition of tin in each type of bronze.

In practical terms, accurate weight computation is essential for a variety of applications, including material science, cooking, and more. In the provided solution, we first determine the total tin content before and after mixing the alloys and then solve for 'x,' which represents the weight of the 4% tin bronze that needs to be added. Mastering weight computations is vital in fields such as chemistry and engineering, where precise mixture ratios can determine the quality of an end product.