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Chapter 1: Problem 51

The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to \(53 \mathrm{~cm}\), and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of 9 cubits and a diameter of 2 cubits. For the stated range, what are the lower value and the upper value, respectively, for (a) the cylinder's length in meters, (b) the cylinder's length in millimeters, and (c) the cylinder's volume in cubic meters?

Short Answer

Expert verified
(a) 3.87 to 4.77 meters, (b) 3870 to 4770 millimeters, (c) 0.717 to 1.337 cubic meters.

Step by step solution

01

Convert Range of Cubit to Meters

First, convert the range of a cubit (43 cm to 53 cm) to meters. Since 1 meter equals 100 cm, we divide both ends of the range:Lower bound: \( \frac{43}{100} = 0.43 \) metersUpper bound: \( \frac{53}{100} = 0.53 \) meters.
02

Calculate Cylinder's Length in Meters

The length of the cylinder is given as 9 cubits. To find the lower and upper values of the cylinder's length in meters, multiply the number of cubits by the cubit in meters.Lower value: \( 9 \times 0.43 = 3.87 \) metersUpper value: \( 9 \times 0.53 = 4.77 \) meters.
03

Calculate Cylinder's Length in Millimeters

Convert the results from meters to millimeters by multiplying by 1000 (since 1 meter = 1000 mm).Lower value: \( 3.87 \times 1000 = 3870 \) mmUpper value: \( 4.77 \times 1000 = 4770 \) mm.
04

Calculate Cylinder's Volume in Cubic Meters

The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. The radius is half the diameter.- Lower radius: \( 0.43 \times 2 / 2 = 0.43 \) meters; Lower height (Length): \( 3.87 \) meters. Lower volume: \( V = \pi \times (0.43)^2 \times 3.87 \approx 0.717 \) cubic meters.- Upper radius: \( 0.53 \times 2 / 2 = 0.53 \) meters; Upper height (Length): \( 4.77 \) meters. Upper volume: \( V = \pi \times (0.53)^2 \times 4.77 \approx 1.337 \) cubic meters.

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Key Concepts

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

Cylinder Volume
The volume of a cylinder is a distinct geometric concept involving its height and base area, defined by the formula \[V = \pi \times r^2 \times h\]where \( V \) is the volume, \( \pi \) is a mathematical constant approximately equal to 3.14159, \( r \) is the radius of the cylinder's base, and \( h \) is the cylinder's height.
- To find the cylinder's volume, first calculate the base area, which is the circle at the cylinder's end. This requires squaring the radius and multiplying it by \( \pi \).- Multiply this base area by the height of the cylinder to find the total volume.In our exercise, the cylinder's diameter is provided, so you must halve it to find the radius. Calculating with both the lower and upper bounds of the cubit ensures an understanding of measurement variations, ultimately affecting the cylinder's final volume.
Unit Conversion
Unit conversion is the process of changing the measurement of a quantity from one unit to another. This is crucial for understanding historical measurement systems, such as converting lengths and volumes in ancient units like cubits into modern metrics such as meters and millimeters.- To convert from centimeters to meters, divide the number of centimeters by 100 because 1 meter equals 100 centimeters. For example, a length of 43 cm is equivalent to \( 0.43 \) meters.- Conversion from meters to millimeters calls for multiplying by 1000, since a meter contains 1000 millimeters.Correct unit conversion helps in performing accurate calculations and is vital when dealing with measurements in varying units, such as converting cubit lengths to calculate the cylindrical volume in contemporary measurements.
Measurement Accuracy
Measurement accuracy refers to the closeness of a measured value to a standard or known value. When using ancient units like the cubit, which varies between 43 cm and 53 cm, measurement accuracy becomes tricky due to this natural variability. - In calculations involving ancient units, it is vital to consider both the minimum and maximum values in conversions and calculations to account for inherent inaccuracies and variances in the original measurements. - Defining a range for measurements provides a buffer that accounts for discrepancies; this is particularly useful in historical contexts where measurement tools may not have been standardized. The process of accounting for lower and upper measurements enables a more comprehensive comprehension of possible values, enhancing the accuracy of modern interpretation of historical data.

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Most popular questions from this chapter

The fastest growing plant on record is a Hesperoyucca whipplei that grew \(3.7 \mathrm{~m}\) in 14 days. What was its growth rate in micrometers per second?

In the United States, a doll house has the scale of \(1: 12\) of a real house (that is, each length of the doll house is \(\frac{1}{12}\) that of the real house) and a miniature house (a doll house to fit within a doll house) has the scale of \(1: 144\) of a real house. Suppose a real house (Fig. 1-7) has a front length of \(20 \mathrm{~m}\), a depth of \(12 \mathrm{~m}\), a height of \(6.0 \mathrm{~m}\), and a standard sloped roof (vertical triangular faces on the ends) of height \(3.0 \mathrm{~m}\). In cubic meters, what are the volumes of the corresponding (a) doll house and (b) miniature house?

A ton is a measure of volume frequently used in shipping, but that use requires some care because there are at least three types of tons: A displacement ton is equal to 7 barrels bulk, a freight ton is equal to 8 barrels bulk, and a register ton is equal to 20 barrels bulk. A barrel bulk is another measure of volume: 1 barrel bulk \(=0.1415 \mathrm{~m}^{3}\). Suppose you spot a shipping order for "73 tons" of M\&M candies, and you are certain that the client who sent the order intended "ton" to refer to volume (instead of weight or mass, as discussed in Chapter 5 ). If the client actually meant displacement tons, how many extra U.S. bushels of the candies will you erroneously ship if you interpret the order as (a) 73 freight tons and (b) 73 register tons? \(\left(1 \mathrm{~m}^{3}=28.378 \quad\right.\) U.S. bushels.)

In purchasing food for a political rally, you erroneously order shucked medium-size Pacific oysters (which come 8 to 12 per U.S. pint) instead of shucked medium-size Atlantic oysters (which come 26 to 38 per U.S. pint). The filled oyster container shipped to you has the interior measure of \(1.0 \mathrm{~m} \times 12 \mathrm{~cm} \times 20 \mathrm{~cm}\), and a U.S. pint is equivalent to \(0.4732\) liter. By how many oysters is the order short of your anticipated count?

A vertical container with base area measuring \(14.0 \mathrm{~cm}\) by \(17.0 \mathrm{~cm}\) is being filled with identical pieces of candy, each with a volume of \(50.0 \mathrm{~mm}^{3}\) and a mass of \(0.0200 \mathrm{~g}\). Assume that the volume of the empty spaces between the candies is negligible. If the height of the candies in the container increases at the rate of \(0.250 \mathrm{~cm} / \mathrm{s}\), at what rate (kilograms per minute) does the mass of the candies in the container increase?
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