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Chapter 0: Problem 36

Find the midpoint of the given interval. $$ [7.3,12.7] $$

Short Answer

Expert verified
The midpoint of the interval is \(10\).

Step by step solution

01

Identify the endpoints of the interval

The endpoints of the interval are \(7.3\) and \(12.7\).
02

Calculate the midpoint

To find a midpoint, add the two endpoints and divide by 2. Therefore, the midpoint of the interval would be \((7.3+12.7) / 2\).

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Key Concepts

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

Intervals in Math
Intervals in math describe a set of numbers lying between two specific numbers, known as endpoints. These endpoints define the interval's boundaries. There are different types of intervals depending on how they include these endpoints:
  • Open Interval: Both endpoints are excluded, written as \((a, b)\).
  • Closed Interval: Both endpoints are included, written as \([a, b]\).
  • Semi-open or Semi-closed Interval: Includes one endpoint while excluding the other, written as \((a, b]\) or \([a, b)\).
In the exercise provided, the interval \([7.3, 12.7]\) is a closed interval, meaning it includes both endpoints. Understanding intervals helps in many mathematical operations like finding midpoints, ranges, and more.
Arithmetic Mean
The arithmetic mean, often known as the average, is a critical concept for finding midpoints. It is calculated by summing up a set of numbers and then dividing by the count of those numbers. When dealing with two numbers, such as the endpoints of an interval, the arithmetic mean is used to find the midpoint.

For example, the arithmetic mean of two endpoints \(a = 7.3\) and \(b = 12.7\) is calculated as:
\[ \text{Midpoint} = \frac{(a + b)}{2} \]
This formula simplifies to:
\[ \frac{(7.3 + 12.7)}{2} = 10 \]
This midpoint represents the exact middle of the interval and is a fundamental part of various statistical and mathematical assessments.
Endpoints of an Interval
Endpoints are crucial in defining intervals. They mark the starting and ending points of an interval on a number line. In the interval \([7.3, 12.7]\), the endpoints are 7.3 and 12.7.

  • Endpoints can be included in the interval, indicating a closed interval.
  • If endpoints are not included, it is referred to as an open interval.
Endpoints not only help in identifying the boundaries of an interval but are also essential in calculations such as finding midpoints and determining the length of an interval (found by subtracting the lower endpoint from the upper endpoint). Recognizing how endpoints are defined helps in understanding the nature and type of the interval you are handling.

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