91Ó°ÊÓ

As a contrast between the old and the modern and between the large and the small, consider the following: In old rural England 1 hide (between 100 and 120 acres) was the area of land needed to sustain one family with a single plough for one year. (An area of 1 acre is equal to \(4047 \mathrm{~m}^{2}\).) Also, 1 wapentake was the area of land needed by 100 such families. In quantum physics, the cross-sectional area of a nucleus (defined in terms of the chance of a particle hitting and being absorbed by it) is measured in units of barns, where 1 barn is \(1 \times 10^{-28} \mathrm{~m}^{2} .\) (In nuclear physics jargon, if a nucleus is "largc," then shooting a particle at it is like shooting a bullet at a barn door, which can hardly be missed.) What is the ratio of 25 wapentakes to 11 barns?

Step by step solution

01

- Understand the Problem

We need to find the ratio of the area of 25 wapentakes to the area of 11 barns. To do this, we need to convert both areas to the same unit of measurement (square meters).

02

- Convert Acres to Square Meters

First, find the total number of acres in one wapentake by multiplying the number of acres in one hide by 100 (since one wapentake is the area needed by 100 families). Then, convert the total number of acres to square meters using the conversion factor: \(1 \text{ acre} = 4047 \text{ m}^2\).

03

- Calculate the Area of 1 Wapentake

Assuming the average area of one hide is 110 acres (the midpoint of 100 and 120 acres), the area of one wapentake is \(110 \text{ acres} \times 100 = 11000 \text{ acres}\). Convert this to square meters: \(11000 \text{ acres} \times 4047 \text{ m}^2/\text{acre} = 4.4517 \times 10^7 \text{ m}^2\).

04

- Calculate the Area of 25 Wapentakes

Multiply the area of one wapentake by 25: \(25 \times 4.4517 \times 10^7 \text{ m}^2 = 1.1129 \times 10^9 \text{ m}^2\).

05

- Calculate the Area of 11 Barns

Since 1 barn is equal to \(1 \times 10^{-28} \text{ m}^2\), the area of 11 barns is \(11 \times 1 \times 10^{-28} \text{ m}^2 = 1.1 \times 10^{-27} \text{ m}^2\).

06

- Find the Ratio

Now, calculate the ratio of the area of 25 wapentakes to the area of 11 barns: \[ \frac{1.1129 \times 10^9 \text{ m}^2}{1.1 \times 10^{-27} \text{ m}^2} = 1.0117 \times 10^{36} \].

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion

Unit conversion is essential in physics problem-solving. It allows you to compare quantities accurately. In our exercise, we need to convert acres to square meters and barns to square meters. This makes it possible to find the ratio between areas.

To start, you need some conversion factors: 1 acre equals 4047 square meters, and 1 barn equals 1\(\times\)10\(^{-28}\) square meters. These factors help convert your given areas into the same unit.

Quick tips for unit conversion:

• Identify the units you have and the units you need.
• Use the correct conversion factor.
• Multiply or divide as necessary to get the units you want.

In this exercise, you multiply the number of acres by 4047 to get square meters. For barns, the process is simpler because you're already given the conversion factor directly to square meters.

Area Calculation

Calculating area is crucial when comparing land in different measurements. Here, we calculate the area of one wapentake and then 25 wapentakes.

Steps:

• Determine the area of one hide, using the average value of 110 acres.
• Multiply by 100 (the number of hides per wapentake) to get 11000 acres.
• Convert this area into square meters: 11000 acres \(\times\) 4047 \(\text{m}^2/\text{acre}\) = 4.4517 \(\times\) 10\(^{7}\) \(\text{m}^2\).
• Multiply to find the area of 25 wapentakes: 25 \(\times\) 4.4517 \(\times\) 10\(^{7}\) \(\text{m}^2\) = 1.1129 \(\times\) 10\(^{9}\) \(\text{m}^2\).

This approach ensures that you accurately measure large and small areas, making future calculations simpler.

Ratio Calculation

Once you have the areas converted to the same unit, calculating the ratio becomes straightforward.

Steps to calculate the ratio:

• Identify both areas in the same unit (square meters in this case).
• Use the formula for ratio: \( \text{Ratio} = \frac{\text{Area of 25 wapentakes}}{\text{Area of 11 barns}} \).
• Plug in the numbers: \( \frac{1.1129 \times 10^{9} \ \text{m}^2}{1.1 \times 10^{-27} \ \text{m}^2} = 1.0117 \times 10^{36} \)

Ratios let you compare two quantities directly. In this case, the huge ratio shows how massively bigger the area of 25 wapentakes is compared to 11 barns. Understanding ratios helps in grasping the scale of differences between various quantities.