Problem 11 - Technical Mathematics with Calculus

Question

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A concrete mixture is to be made which contains \(35 \%\) sand by weight, and 642 lb of mixture containing \(29 \%\) sand is already on hand. How many pounds of sand must be added to this mixture to arrive at the required \(35 \% ?\)

Step-by-Step Solution

Answer

186 lb of sand must be added.

Step 1: Calculate the Amount of Sand in the Existing Mixture

Determine the amount of sand already in the mixture by calculating 29% of 642 lb. Use the percentage formula: amount of sand = (percentage of sand / 100) * total weight of mixture.

Step 2: Set Up an Equation for the Desired Mixture

Let the amount of sand that needs to be added be represented by x. The total weight of the new mixture will be 642 lb + x. The desired mixture needs to have 35% sand, so set up the equation: (amount of sand in the existing mixture + x) / (642 lb + x) = 35%.

Step 3: Solve for x

Plug in the amount of sand from step 1 into the equation and solve for x, which represents the amount of sand to be added. Multiply each side of the equation by (642 + x) to eliminate the fraction, and re-arrange the equation to solve for x.

Step 4: Perform the Calculations

Perform the calculations according to the algebraic steps outlined previously, and round the result to three significant digits.

Key Concepts

Percentage Formula

The percentage formula is a mathematical way to express a number as a fraction of 100. It's widely used to compare quantities and to identify ratios in different substances or components. For instance, if you have a concrete mixture with a certain percentage of sand, you can determine how much sand is present or needed by using this formula:

• To calculate the percentage of a specific component: \(\text{percentage of component} = \frac{\text{amount of component}}{\text{total}} \times 100\)
• To find the amount from a known percentage: \(\text{amount of component} = \frac{\text{percentage of component}}{100} \times \text{total}\)

In our exercise, to understand the current amount of sand in the mixture, we determine 29% of 642 lb using the second formula, which elucidates the interconnection between percentages and weight calculations in practical situations.

Significant Digits

Significant digits, or significant figures, are essential in science and engineering to indicate the precision of measurements and computations. When calculating with percentages, especially in chemistry or material science problems such as mixing concrete, it's important to consider the right number of significant figures. It helps in maintaining the accuracy of a calculation throughout a series of computations.For instance, when dealing with percentages as exact numbers, we calculate to a defined number of significant digits (in our case, three). This ensures that the precision reflects the reliability of the input data. When rounding off the final result or intermediate steps, it's crucial to keep to the three significant digits to avoid errors due to overapproximation or underestimation. This principle ensures that our calculations yield reliable and practical results in real-world applications.

Algebraic Equation Solving

Algebraic equation solving is at the heart of figuring out problems involving unknown quantities. This skill aids in managing equations to find a variable, like the amount of a substance in a mixture. Here's what you typically do:

• Identify the unknown variable (let's call it 'x').
• Create an equation based on the relationship between the known and unknown quantities.
• Rearrange the equation so that x is on one side, making it the subject.
• Simplify and solve the equation.

In our exercise, we set up an equation to find the amount of sand to be added ('x') to arrive at the desired mixture of 35% sand. The solving process may include various steps—like multiplying both sides to eliminate fractions, and isolating x—which requires careful algebraic manipulation. Mastering these techniques allows for accurate and efficient problem-solving.

Weight Calculation

Weight calculation is a significant aspect of many scientific and mathematical problems, especially when it comes to mixing substances to achieve a certain composition. In our problem, it involves determining how much sand weighs in a mixture and how much needs to be added.

Current Mixture Weight Calculation

First, we calculate the weight of sand in the existing mixture by finding 29% of 642 lb. This involves applying the percentage formula mentioned earlier.

Additional Weight Calculation

After determining the weight of sand already present, we set up an algebraic equation to compute the additional weight of sand needed to reach the desired 35% composition. At this point, weight calculations intertwine with algebraic equation solving, demonstrating the interdisciplinary nature of such problems. By accurately performing weight calculations in combination with the right percentage and algebraic methods, we can manipulate and solve complex, practical situations like mixing materials to specific requirements.