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Chapter 1: Problem 30

A gray is an old English measure for length, defined as \(1 / 10\) of a line, where line is another old English measure for length, defined as \(1 / 12\) inch. A common measure for length in the publishing business is a point, defined as \(1 / 72\) inch. What is an area of \(0.50\) gry \(^{2}\) in terms of points squared (points \(^{2}\) )?

Short Answer

Expert verified
The area of 0.50 gry^2 in terms of points^2 is 200 points^2.

Step by step solution

01

Understand the Conversion from Grays to Lines

Recognize that 1 gray (gry) is equal to \(\frac{1}{10}\) of a line.
02

Understand the Conversion from Lines to Inches

Realize that 1 line is equal to \(\frac{1}{12}\) of an inch.
03

Convert Grays to Inches

Since \(1 \text{gry} = \frac{1}{10} \text{line}\) and \(1 \text{line} = \frac{1}{12} \text{inch}\), combine these conversions to get: \[1 \text{gry} = \frac{1}{10} \times \frac{1}{12} = \frac{1}{120} \text{inch}\] Therefore, \[0.50 \text{gry}^2 =0.50 \text{gry}\times 0.50 \text{gry} = 0.50^2 \times \left( \frac{1}{120} \text{inch} \right)^2 \]
04

Convert Inches to Points

Recognize that 1 point is equal to \(\frac{1}{72}\) inch. Then convert the grays squared to points squared using the conversion \(1 \text{inch} = 72 \text{points}\): \[ ( \frac{1}{120} \text{inch} )^2 = \frac{1}{14400} \text{inch}^2\]Finally,\[ 0.25 \times \left( \frac{1}{14400} \right) = \frac{1}{72 \times 72} ( \text{points} )\]
05

Calculate the Final Area

Multiply the final conversion from inch squared to points squared \[ \text{inch}^2 \to points^2 \]Calculate as follows: \[0.25\times \frac{1}{14400} \times (72 \text{points})^2 = \frac{0.25}{200} = 200 points^2\]

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Key Concepts

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

grays and lines conversion
Grays and lines are old English measures for length. One gray (gry) is defined as \(\frac{1}{10}\) of a line. This conversion is straightforward: for every line, you have ten grays. Let's break this down:
  • 1 gray (gry) = \(\frac{1}{10}\) of a line
So, if you know the length in lines, simply multiply by ten to convert to grays. This concept is fundamental, as it sets the basis for further conversions, such as converting to inches.
lines to inches conversion
The next step involves converting lines to inches, another critical conversion in this series. In old English measurements, one line is equal to \(\frac{1}{12}\) of an inch. This means that for every inch, there are twelve lines. Here is how you represent this:
  • 1 line = \(\frac{1}{12}\) inch
Combining the previous and current conversions, we get:
  • 1 gray (gry) = \(\frac{1}{10} \times \frac{1}{12} = \frac{1}{120} \) inch
Now you can understand how small a gray is when expressed in inches.
linear to area conversion
To convert a linear measurement to an area measurement, you need to square the linear dimensions. For instance, if you have 0.50 gry2, first understand it in linear terms before squaring each part:
  • 1 gry = \(\frac{1}{120}\) inch
Therefore, an area calculation involves:
  • 0.50 gry = \(\frac{0.50}{120}\) inch = \(\frac{1}{240}\) inch
  • Area (0.50 gry2) = \(\frac{1}{240} \times \frac{1}{240} = \frac{1}{14400}\) square inches
This approach lays the groundwork for transitioning from linear to area measurements.
conversion from inches to points
Points are a common measure in publishing, where 1 point equals \(\frac{1}{72}\) of an inch. To convert from square inches to points squared, use:
  • 1 inch = 72 points
  • 1 inch2 = (72 points)2 = 5184 points2
Now if you have \(\frac{1}{14400}\) square inches, convert this using the above:
  • Area = \(\frac{1}{14400} \times 5184 = \frac{0.25}{200} = 200\) points2
Thus, the final answer is 200 points squared.

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Most popular questions from this chapter

Historically the English had a doubling system when measuring volumes; 2 mouthfuls equal 1 jigger, 2 jiggers equal 1 jack (also called a jackpot); 2 jacks equal 1 jill; 2 jills \(=\) 1 cup; 2 cups \(=1\) pint; 2 pints \(=1\) quart; 2 quarts \(=1\) pottle; 2 pottles \(=1\) gallon; 2 gallons \(=1\) pail. (The nursery rhyme "Jack and Jill" refers to these units and was a protest against King Charles I of England for his taxes on the jacks of liquor sold in the tavern. (See A. Kline, The World of Measurement, New York: Simon and Schuster, 1975, pp. \(32-39 .\) American and British cooks today use teaspoons, tablespoons, and cups; 3 teaspoons \(=1\) tablespoon; 4 tablespoons \(=1 / 4\) cup. Assume that you find an old English recipe requiring 3 jiggers of milk. How many cups does this represent? How many tablespoons? You can assume that the cups in the two systems represent the same volume.

Hydraulic engineers in the United States often use, as a unit of volume of water, the acre-foot, defined as the volume of water that will cover 1 acre of land to a depth of \(1 \mathrm{ft}\). A severe thunderstorm dumped \(2.0\) in. of rain in \(30 \mathrm{~min}\) on a town of area \(26 \mathrm{~km}^{2}\). What volume of water, in acre-feet, fell on the town?

Here are two related problems - one precise, one an estimation. (a) A sculptor builds model for a statue 0 . a terrapin to replace Testudo." She discov. ers that to cast her small scale model she needs \(2 \mathrm{~kg}\) of bronze When she is done, she finds that she can give it two coats of finish. ing polyurethane varnish using exactly one small can of varnish. The final statue is supposed to be 5 times as large as the model in cach dimension. How much bronze will she need? How much varnish should she buy? (Hint: If this seems difficult, you might start by writing a simpler question that is easier to work on before tackling this one.) (b) The human brain has 1000 times the surface area of a mouse's brain. The human brain is convoluted, the mouse's is not. How much of this factor is due just to size (the human brain is bigger)? How sensitive is your result to your estimations of the approximate dimensions of a human brain and a mouse brain?

A unit of time sometimes used in microscopic physics is the shake. One shake equals \(10^{-8} \mathrm{~s}\). (a) Are there more shakes in a second than there are seconds in a year? (b) Humans have existed for about \(10^{6}\) years, whereas the universe is about \(10^{10}\) years old. If the age of the universe now is taken to be 1 "universe day," for how many "universe seconds" have humans existed?

Discuss the question: "Is 500 feet big or small?" Before you do so, carry out the following estimates. (a) You are on the top floor of a 500 -ft-tall building. A fire breaks out in the building and the elevator stops working. You have to walk down to the ground floor. Estimate how long this would take you. (Your stairwell is on the other side of the building from the fire.) (b) You are hiking the Appalachian Trail on a beautiful fall morning as part of a \(10 \mathrm{mi}\) hike with a group of friends. You are walking along a well-tended, level part of the trail. Estimate how long it would take you to walk \(500 \mathrm{ft}\). (c) You are driving on the New Jersey Turnpike at \(65 \mathrm{mi} / \mathrm{hr}\). You pass a sign that says "Lane ends 500 feet." How much time do you have in order to changelanes?
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