Problem 65 - Chemistry The Central Science

Question

The US 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 US National Debt Clock showed the outstanding public debt to be $$\$ 11,687,233,914,811.11$$ on August \(19,2009 .\) How many stacks like the one described would be necessary to pay off this debt?

Step-by-Step Solution

Answer

To answer the given question: (a) Approximately 112,501 quarters would need to be stacked to reach the height of the Washington Monument. (b) The total weight of this stack would be about 1,468.37 pounds. (c) This stack would have a total value of \(\$28,125.25\). (d) To pay off the debt, about 415,492 stacks like the one described would be required.

Step 1: Calculate the number of quarters needed to reach 575 ft

First, we need to convert the thickness of a quarter, which is given in mm, to ft: \(1.55~mm * (1~in/(25.4~mm))(1~ft/(12~in))\) Now, we divide the height of the Washington Monument (575 ft) by the calculated thickness in ft to find the number of quarters needed. \(Number~of~Quarters = \frac{Height~of~Washington~Monument}{Thickness~of~a~Quarter~in~ft}\)

Step 2: Calculate the total weight of the stack

Since the mass of one quarter is given in grams, we will convert it to pounds by using the conversion factor. Then, we'll multiply the number of quarters from Step 1 by the mass of one quarter to find the total weight of the stack. \(5.67~g * (1~lb)/(453.592~g)\) \(Total~Weight = Number~of~Quarters * Weight~of~a~Quarte_r~in~lb\)

Step 3: Calculate the total value of the stack

Each quarter is worth \(0.25. We'll multiply the number of quarters from Step 1 by \)0.25 to find the total value of the stack. \(Total~Value = Number~of~Quarters * 0.25\)

Step 4: Calculate the number of stacks required to pay off the debt

We'll divide the outstanding public debt by the total value of one stack (calculated in Step 3) to find the number of stacks required to pay off the public debt. \(Number~of~Stacks = \frac{Outstanding~Public~Debt}{Total~Value~of~One~Stack}\)

Key Concepts

Metric System Conversions

Understanding metric system conversions is essential, especially when working with measurements in different units. In the exercise involving the US quarter and the Washington Monument, we begin by converting the thickness of a quarter from millimeters to feet. Since 1 inch equals 25.4 millimeters, and 1 foot equals 12 inches, we need to perform a two-step conversion:

• Convert millimeters to inches: Multiply the value in mm by the conversion factor (1 in/25.4 mm).
• Then, convert inches to feet by using (1 ft/12 in).

This conversion helps in determining how many quarters stacked on top of each other would equal the height of the Washington Monument (575 feet). Conversions like these are common in scientific problems where measurements may initially be presented in less convenient units for the task at hand. Breaking down complex problems into a series of straightforward conversions simplifies the process.

Unit Conversion

Unit conversion is a crucial skill in problem-solving, especially within the realm of chemistry and physics. In this exercise, we encounter the need to convert grams to pounds to find the total weight of a stack of quarters. The conversion from grams to pounds uses the relationship: 453.592 grams equals 1 pound. To execute the conversion:

• Multiply the mass in grams by the conversion factor (1 lb/453.592 g).
• This gives you the weight in pounds, useful when discussing mass in the context of larger weights or when following a consistent unit system throughout the problem.

Accurate unit conversion ensures that calculations maintain a high level of precision and are understandable, since consistent units make it easier to compare and analyze data.

Problem Solving in Chemistry

Problem-solving in chemistry often requires methodical and logical thinking, as shown in the exercise about stacking quarters. This exercise demonstrates several analytical steps common in chemistry problem-solving. You are tasked with stacking quarters to a certain height, weighing them, calculating their value, and then determining how many such stacks would be needed to match a certain debt. Key steps involve:

• Identifying the objective: For each sub-question, identify the end goal, which might be height, weight, or monetary value.
• Breaking down the task: Convert measurements, calculate per unit mass and value, and finally, apply these to find cumulative totals.
• Using calculations iteratively: Move step-by-step from the basic input information to final insights, ensuring each calculation is clearly justified by the problem's requirements.

This logical breakdown of tasks is essential in tackling larger chemistry problems, providing a clear path from hypothesis through to conclusion.