Problem 69 - Chemistry: The Central Science

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

(a) We would need \(11,302,690\) quarters to be stacked to reach the height of the Washington Monument. (b) The stack would weigh \(64,097,272.3 \mathrm{~g}\). (c) The stack would contain \(\$2,825,672.50\). (d) We would need \(5,732,511\) stacks to pay off the US National Debt as of October 28, 2012.

Step 1: Find the number of quarters

To find the number of quarters that need to be stacked to reach the height of the Washington Monument, we first need to convert the height into the same unit as the thickness of the quarter (millimeters). The height of the Washington Monument is 575 feet. We will need to convert it to millimeters: $$1 \, ft = 30.48\, cm$$ $$1\, cm = 10\, mm$$Using these conversions, $$575\, ft = 17,51620\, mm$$Now we can find the number of quarters by dividing the height in millimeters by the thickness of one quarter:$$\frac{17,51620\, mm}{1.55\, mm}$$

Step 2: Calculate the Total Weight of the Stack

To find the weight of the stack, simply multiply the number of quarters by the mass of 1 quarter (5.67 g):$$\text{Stack Weight} = \text{Total Quarters} \times \text{Mass of 1 Quarter} = \text{Total Quarters} \times 5.67\, g$$

Step 3: Calculate the Total Value of the Stack

To calculate the total value of the stack, multiply the number of quarters by their face value, which is 25 cents or $0.25:$$\text{Stack Value} = \text{Total Quarters} \times \$0.25$$

Step 4: Determine Stacks Needed to Pay Off Debt

To find how many stacks like the one described are needed to pay off the US National Debt, divide the debt amount by the value of one stack:$$\text{Stacks Needed} = \frac{\$16,213,166,914,811\}{\text{Stack Value}}$$ Now we can plug the values and solve the problem:

Step 5: Plugging the Values and Solving

(a) Number of quarters to reach the height of the Washington Monument:$$\frac{17,51620\, mm}{1.55\, mm} = 11,302,690$$We would need 11,302,690 quarters to be stacked. (b) Weight of the stack:$$\text{Stack Weight} = 11,302,690 \times 5.67\, g = 64,097,272.3\, g$$The stack would weigh 64,097,272.3 grams. (c) Stack value:$$\text{Stack Value} = 11,302,690 \times \$0.25 = \$2,825,672.50$$The stack would contain $2,825,672.50. (d) Stacks needed to pay off the US National Debt:$$\text{Stacks Needed} = \frac{\$16,213,166,914,811\}{\$2,825,672.50} = 5,732,511$$We would need 5,732,511 stacks to pay off the US National Debt as of October 28, 2012.

Key Concepts

Unit Conversion

Unit conversion is a fundamental skill in chemistry that allows measurements in different units to be compared or combined. In this problem, to determine how many quarters stacked together would reach the height of the Washington Monument, we first need to convert the height into millimeters. This aligns the units of the monument's height with the thickness of a quarter (in millimeters).

Start by understanding the conversion factors:

• 1 foot is equivalent to 30.48 centimeters
• 1 centimeter is equivalent to 10 millimeters

By using these unit conversions, you can transform the height of the monument from feet to millimeters, enabling direct comparison with the thickness of quarters. This step underscores the importance of precise unit conversion in ensuring accuracy in scientific problem-solving.

Mathematical Calculations in Chemistry

Mathematical calculations form the backbone of quantitative analysis in chemistry. In our exercise, we first tackle the challenge of aligning the quarter's thickness with the tower's height using division. Calculating the number of quarters to reach the desired height involves dividing the constructed height (in millimeters) by the thickness of one quarter.

Here’s the calculation:

• Divide the total height of the Washington Monument in millimeters by the thickness of a quarter: \( \frac{17,51620 \text{ mm}}{1.55 \text{ mm}} \)

The result gives the sheer number of quarters required. Each subsequent step leverages this crucial figure, whether determining the stack’s weight or its total monetary value, by involving further multiplication with the known mass and monetary value of a single quarter.

Applied Problem-Solving in Chemistry

Applied problem-solving is the real-world application of mathematical and scientific principles to address practical issues. In this chemistry-oriented exercise, you are asked to extrapolate data to evaluate complex scenarios, such as weighing and valuing a vast number of coins or assessing national debt equivalency to a physical stack of quarters.

This entails understanding:

• The mass influenced by the calculated number of quarters: multiply the total number by a quarter’s weight (\(5.67 \text{ g}\)) to get the collective weight.
• The economic impact by equating the total number of quarters to their total nominal value in currency \(\text{(Total Quarters} \times \$0.25)\).
• Complex debt relationships, modeled by testing the realistic implication of quarter stacks against gigantic figures like national debt, using basic division of the debt by the cumulative stack value.

These problems not only practice arithmetic but enhance deeper critical-thinking skills by interpreting and solving tangible, real-world problems.