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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 710 to 1521 oysters.

Step by step solution

01

Calculate Container Volume in Liters

First, we need to calculate the volume of the container in liters. The dimensions of the container are given as 1.0 m by 12 cm by 20 cm, which need to be converted into meters: 1.0 m, 0.12 m, and 0.20 m. The volume in cubic meters is \(1.0 \times 0.12 \times 0.20 = 0.024\) cubic meters. To convert to liters, we multiply by 1000 (since 1 cubic meter = 1000 liters), giving us \(0.024 \times 1000 = 24\) liters.
02

Convert Container Volume to Pints

Next, convert the volume from liters to U.S. pints, since the oyster quantities are given per pint. With the conversion factor 1 U.S. pint = 0.4732 liters, the number of pints in the container is \(\frac{24}{0.4732} \approx 50.72\) pints.
03

Calculate Expected and Actual Oyster Counts

Calculate the number of oysters for each type: - **Atlantic Oysters**: With 26 to 38 oysters per pint and 50.72 pints, the expected oyster count range is \(50.72 \times 26 = 1318.72\) (lower bound) and \(50.72 \times 38 = 1927.36\) (upper bound), rounding to 1319 and 1927 oysters.- **Pacific Oysters**: With 8 to 12 oysters per pint and 50.72 pints, the actual oyster count range is \(50.72 \times 8 = 405.76\) (lower bound) and \(50.72 \times 12 = 608.64\) (upper bound), rounding to 406 and 609 oysters.
04

Calculate Oyster Shortage

Determine the shortage in the number of oysters. The minimum shortage using lower bounds is \(1319 - 609 = 710\) oysters. The maximum shortage using upper bounds is \(1927 - 406 = 1521\) oysters. Thus, the order is short by at least 710 oysters and at most 1521 oysters.

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Key Concepts

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

Unit Conversion
When solving problems involving volume, unit conversion is a crucial step. It's important to understand how to convert between different units of measurement to ensure accuracy.
In our exercise, we begin by converting the dimensions of the container from centimeters to meters to align with the unit system used for volume calculation.
Here鈥檚 how you typically perform these conversions:
  • Identify the units you need to convert from and to, like centimeters to meters.
  • Use the conversion factor, for centimeters to meters, this is 0.01 (since 1 cm = 0.01 m).
  • Apply this factor to the given measurements. For example, 12 cm becomes 0.12 m.
Furthermore, when converting from cubic meters to liters, it鈥檚 vital to remember:
  • 1 cubic meter contains 1000 liters.
  • For example, multiplying 0.024 cubic meters by 1000 gives 24 liters.
Understanding these conversions aids in the seamless progression to estimating the necessary quantities effectively.
Mathematical Problem Solving
Engaging in mathematical problem solving involves breaking down the problem into manageable parts and using logical reasoning to reach a solution.
This exercise involves several steps that demonstrate this skill.
  • First, identify what the problem is asking 鈥� like how many oysters are short in the order.
  • Then, gather the data, which includes knowing the number of oysters per pint and the conversion factors between units (e.g., liters to pints).
  • Next, make calculations step-by-step such as finding the total volume in liters, converting liters to pints, and calculating the total number of oysters expected and received.
  • Finally, compare the expected number of oysters to the actual number received to determine the shortage.
By following these logical steps, the problem is simplified, allowing for clear resolution.
Dimensional Analysis
Dimensional analysis is a technique used in mathematical problem solving to ensure that all calculations maintain consistency across units.
In volume-related problems, this is particularly important. Here's how it is applied in the exercise:
  • Start with ensuring all measurements are consistent. Convert all dimensions to a single unit system (meters or centimeters) before calculating volume.
  • Once converted, use these measurements to determine the container's volume accurately, which was calculated to be 24 liters in this exercise.
  • Dimensional analysis then ensures the final figure is logical when converting from one quantity to another, such as liters to pints, where the conversion factor of 0.4732 liters per U.S. pint is employed.
  • In each step, check if the units correctly cancel out or convert to the desired outcome.
This approach prevents errors and guarantees accurate interpretation of results, a critical aspect for solving complex, multi-step problems.

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Most popular questions from this chapter

You are to fix dinners for 400 people at a convention of Mexican food fans. Your recipe calls for 2 jalape帽o peppers per serving (one serving per person). However, you have only habanero peppers on hand. The spiciness of peppers is measured in terms of the scoville heat unit (SHU). On average, one jalape帽o pepper has a spiciness of \(4000 \mathrm{SHU}\) and one habanero pepper has a spiciness of 300000 SHU. To get the desired spiciness, how many habanero peppers should you substitute for the jalape帽o peppers in the recipe for the 400 dinners?

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