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Chapter 1: Problem 59

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?

Short Answer

Expert verified
The order is short by 709 to 1521 oysters.

Step by step solution

01

Convert Container Volume to Liters

The container has dimensions of \(1.0 \, \text{m} \times 12 \, \text{cm} \times 20 \, \text{cm}\). First, convert the dimensions from centimeters to meters: \(12 \, \text{cm} = 0.12 \, \text{m}\) and \(20 \, \text{cm} = 0.20 \, \text{m}\). Now, calculate the volume in cubic meters: \(1.0 \, \text{m} \times 0.12 \, \text{m} \times 0.20 \, \text{m} = 0.024 \, \text{m}^3\). Then convert to liters: \(1 \, \text{m}^3 = 1000 \, \text{L}\), so the container has \(24 \, \text{L}\).
02

Convert Volume to U.S. Pints

A U.S. pint is equivalent to \(0.4732 \, \text{L}\). To find how many pints fit in \(24 \, \text{L}\), divide the volume by the volume of one pint: \(\frac{24}{0.4732} \approx 50.71\) pints.
03

Calculate Expected and Actual Oyster Quantities

For Atlantic oysters, the expected order ranges from \(26\) to \(38\) oysters per pint. Calculate the total expected amount: \(50.71 \times 26 = 1318.46\) and \(50.71 \times 38 = 1926.98\). Thus, you expected between \(1318\) and \(1927\) oysters. For Pacific oysters, the actual order ranges from \(8\) to \(12\) oysters per pint. Calculate the total actual amount: \(50.71 \times 8 = 405.68\) and \(50.71 \times 12 = 608.52\). Thus, you actually received between \(406\) and \(609\) oysters.
04

Calculate the Shortfall Range

Find the shortfall by subtracting the number of actual oysters from the expected count for each range: \(1318 - 609 = 709\) and \(1927 - 406 = 1521\). Thus, the order is short by between \(709\) and \(1521\) oysters.

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Key Concepts

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

Unit Conversion
Unit conversion is a fundamental mathematical skill that allows us to change measurements from one unit to another. In many real-world scenarios, like calculating the volume of a container, converting units helps in understanding quantities in a more familiar metric. In the case of the oyster container, we began with dimensions given in meters and centimeters.
  • First, it's crucial to understand that a meter is the base unit of length in the metric system, while the centimeter is a smaller unit. There are 100 centimeters in a meter.
  • When converting, change centimeters to meters by dividing by 100. For example, 12 cm becomes 0.12 meters, and 20 cm becomes 0.20 meters.
  • With all dimensions in the same unit, calculate the container's volume in cubic meters and convert that to liters, knowing that 1 cubic meter equals 1000 liters.
This process ensures consistency and accuracy in measurement, simplifying further calculations.
Container Volume Calculation
Calculating the volume of a container involves finding the space inside it, a task often necessary in practical scenarios such as shipping. With the given dimensions, the volume is the product of length, width, and height:
  • Using dimensions in meters as mentioned (1.0 m, 0.12 m, 0.20 m), the formula to calculate the volume is: \[\text{Volume} = \text{length} \times \text{width} \times \text{height} = 1.0 \times 0.12 \times 0.20 = 0.024 \text{ m}^3\]
  • Converting this volume to liters requires knowing the conversion factor: there are 1000 liters in a cubic meter. Thus, the container鈥檚 volume is 24 liters.
Understanding these calculations is essential for determining how much a container can hold, particularly when dealing with liquid or granular products.
Mathematical Problem Solving
Mathematical problem solving combines various skills, including logical reasoning, calculation, and analysis. Here, it guides us through comparing expectations with reality in the context of ordering oysters:
  • First, calculate how many pints fit into the volume of the container by dividing the total volume in liters by the volume of one U.S. pint:\[\text{Pints} = \frac{24}{0.4732} \approx 50.71\]
  • We then calculate the expected and actual quantities of oysters based on given ranges per pint. For instance, anticipate between 1318 and 1927 Atlantic oysters and receive between 406 and 609 Pacific oysters.
  • The shortfall calculation shows the difference, providing a range of how many fewer oysters were shipped than expected: from a minimum of 709 to a maximum of 1521 oysters.Understanding these computations helps manage and verify orders effectively, ensuring efficient operations.

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Most popular questions from this chapter

The common Eastern mole, a mammal, typically has a mass of \(75 \mathrm{~g}\), which corresponds to about \(7.5\) moles of atoms. (A mole of atoms is \(6.02 \times 10^{23}\) atoms.) In atomic mass units (u), what is the average mass of the atoms in the common Eastern mole?

An old manuscript reveals that a landowner in the time of King Arthur held \(3.00\) acres of plowed land plus a livestock area of \(25.0\) perches by \(4.00\) perches. What was the total area in (a) the old unit of roods and (b) the more modern unit of square meters? Here, 1 acre is an area of 40 perches by 4 perches, 1 rood is an area of 40 perches by 1 perch, and 1 perch is the length \(16.5 \mathrm{ft}\).

The record for the largest glass bottle was set in 1992 by a team in Millville, New Jersey-they blew a bottle with a volume of 193 U.S. fluid gallons. (a) How much short of \(1.0\) million cubic centimeters is that? (b) If the bottle were filled with water at the leisurely rate of \(1.8 \mathrm{~g} / \mathrm{min}\), how long would the filling take? Water has a density of \(1000 \mathrm{~kg} / \mathrm{m}^{3}\).

You can easily convert common units and measures electronically, but you still should be able to use a conversion table, such as those in Appendix D. Table \(1-6\) is part of a conversion table for a system of volume measures once common in Spain; a volume of 1 fanega is equivalent to \(55.501 \mathrm{dm}^{3}\) (cubic decimeters). To complete the table, what numbers (to three significant figures) should be entered in (a) the cahiz column, (b) the fanega column, (c) the cuartilla column, and (d) the almude column, starting with the top blank? Express \(7.00\) almudes in (e) medios, (f) cahizes, and (g) cubic centimeters \(\left(\mathrm{cm}^{3}\right)\). Table 1-6 Problem 6 $$ \begin{array}{lccccc} \hline & \text { cahiz } & \text { fanega } & \text { cuartilla } & \text { almude } & \text { medio } \\ \hline 1 \text { cahiz }= & 1 & 12 & 48 & 144 & 288 \\ 1 \text { fanega }= & & 1 & 4 & 12 & 24 \\ 1 \text { cuartilla }= & & & 1 & 3 & 6 \\ 1 \text { almude }= & & & & 1 & 2 \\ 1 \text { medio }= & & & & & 1 \\ \hline \end{array} $$

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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