Problem 75 - Chemistry The Central Science In Si Units

Question

The U.S. quarter has a mass of \(5.67 \mathrm{~g}\) and is approximately \(1.55 \mathrm{~mm}\) thick. (a) How many quarters would have to be stacked to reach \(575 \mathrm{ft}\), the height of the Washington Monument? (b) How much would this stack weigh? (c) How much money would this stack contain? (d) The U.S. National Debt Clock showed the outstanding public debt to be \(\$ 16,213,166,914,811\) on October \(28,2012 .\) How many stacks like the one described would be necessary to pay off this debt?

Step-by-Step Solution

Answer

113,033 quarters are needed to reach the height; the stack weighs 641.84 kg, contains $28,258.25, and 573,614,383 stacks are needed to pay off the national debt.

Step 1: Convert the height of the Washington Monument

The height of the Washington Monument is given as 575 ft. We need to convert this height to millimeters because the thickness of a quarter is given in millimeters. Using the conversion factor, 1 ft = 304.8 mm, we have: \[ 575 \text{ ft} \times 304.8 \text{ mm/ft} = 175,260 \text{ mm} \]This is the height in millimeters.

Step 2: Calculate the number of quarters needed to reach the height

To find out how many quarters are required to reach the height of 175,260 mm, we divide this height by the thickness of one quarter, which is 1.55 mm:\[ \frac{175,260 \text{ mm}}{1.55 \text{ mm/quarter}} \approx 113,032.26 \text{ quarters} \]Rounding up, we find that 113,033 quarters are needed because you cannot have a fraction of a quarter.

Step 3: Calculate the weight of the stack of quarters

Each quarter weighs 5.67 grams. Therefore, the total weight of 113,033 quarters is:\[ 113,033 \text{ quarters} \times 5.67 \text{ g/quarter} = 641,840.71 \text{ grams} \]To convert grams to kilograms: \[ \frac{641,840.71 \text{ g}}{1000 \text{ g/kg}} = 641.84 \text{ kg} \]The stack weighs approximately 641.84 kg.

Step 4: Calculate the monetary value of the stack

Each quarter is worth \(0.25. So, the total monetary value of 113,033 quarters is:\[ 113,033 \times 0.25 = 28,258.25 \text{ dollars} \]The stack contains \)28,258.25.

Step 5: Calculate how many stacks would be needed to pay off the national debt

The national debt is $16,213,166,914,811. To find out how many such stacks are needed to cover this amount, divide the debt by the monetary value of one stack: \[ \frac{16,213,166,914,811}{28,258.25} \approx 573,614,383 \]Thus, around 573,614,383 stacks are needed to pay off the debt.

Key Concepts

Unit Conversion

Unit conversion is a key skill in many scientific fields, including chemistry and physics. It allows us to switch between different units of measurement to understand and convey information accurately. For example, in the given problem, the height of the Washington Monument is given in feet, while the thickness of a quarter is in millimeters.

To solve problems involving different units, we need to use conversion factors. A conversion factor is a ratio that expresses how many of one unit are equivalent to another unit. In this exercise, we used the conversion factor 1 ft = 304.8 mm to convert the height from feet to millimeters. This ensures we can directly compare the thickness of each quarter to the total height needed.

When converting units, it's important to carefully set up your calculations to cancel out the units you want to change. This prevents errors and helps maintain consistency across calculations. Practicing unit conversion will enhance your problem-solving skills and allow for greater flexibility when dealing with complex measurements.

Physical Quantities

Physical quantities are any measurable aspects of the universe that we use in scientific calculations. They have both a magnitude and a unit of measurement. Examples include mass, length, time, and temperature. In the exercise, physical quantities such as the mass of a quarter and the height of the Washington Monument are given and utilized.

Understanding physical quantities involves more than just knowing their values; it's about recognizing how they interact. For example, the thickness of a single quarter (1.55 mm) helps us determine how many quarters need to be stacked to reach a specific height. Similarly, the mass of each quarter (5.67 grams) allows us to calculate the total weight of the stack.

Mastering the concept of physical quantities includes knowing which quantities are relevant to your problem and applying them appropriately. This means being able to read a problem, identify the important numbers, and understand their significance. As you improve, you'll be able to tackle more challenging problems with ease by properly interpreting and manipulating these physical quantities.

Problem Solving in Chemistry

Problem solving in chemistry often involves using a systematic approach to break down and analyze complex questions. In the context of our exercise, various steps are employed to tackle different aspects of the problem.

We start with unit conversion to ensure all measurements align. Then, we use these conversions to determine the number of quarters needed for the desired height, calculate the total weight, and assess the financial value of the quarters. Finally, we consider how many such stacks would be required to address the national debt. Each of these steps involves a combination of logical thinking, mathematics, and an understanding of the physical quantities involved.

In chemistry, this methodical approach is crucial as it can be extended to chemical reactions, balancing equations, and understanding molecular interactions. By breaking down a seemingly overwhelming problem into manageable steps, you can tackle chemistry problems more efficiently. Remember that practice is key, and as you refine your skills, chemical problem solving will become much more intuitive.