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Chapter 9: Problem 49

Without using a calculator, explain why \(2^{\pi}\) must be greater than 8 but less than 16

Short Answer

Expert verified
Since \(3 < \pi < 4\), \(8 < 2^{\pi} < 16\).

Step by step solution

01

Understand the given range

We need to show why \( 8 < 2^{\pi} < 16 \). Since \(8\) and \(16\) are powers of \(2\), rewrite them as \(2^3\) and \(2^4\) respectively.
02

Construct the inequality

We now need to show that \(2^3 < 2^{\pi} < 2^4\). This simplifies to finding where \(3 < \pi < 4\).
03

Relate to the value of \(\pi \)

Recognize that the value of \(\pi\) is approximately \(3.14159\). Checking this within the inequality constructed in the previous step: \(3 \lt 3.14159 \lt 4\).
04

Conclude within the constructed inequality

Since \(3 \lt \pi \lt 4\), we have \(2^3 \lt 2^{\pi} \lt 2^4\). Therefore, \(8 \lt 2^{\pi} < 16\).

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Key Concepts

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

Understanding Inequalities
Inequalities are an essential concept in mathematics. They are expressions that describe the relative size or order of two values. For example, the inequality 3 < 5 tells us that 3 is less than 5.
To solve inequalities effectively:
  • Identify the range given in the problem. In this exercise, we needed 8 < 2^蟺 < 16.
  • Recognize patterns. Here, 8 and 16 are both powers of 2, which simplifies rewriting the inequality as 2^3 < 2^蟺 < 2^4.
  • Compare the values. With 蟺 approximately equal to 3.14159, it clearly falls between the whole numbers 3 and 4.
This range allows us to conclude that 2^3, which is 8, is less than 2^蟺, and 2^蟺 is less than 2^4, which is 16. Hence, we showed why 2^蟺 must be greater than 8 but less than 16.
Powers of 2
Powers of 2 are a common topic in math. They refer to numbers that can be expressed as 2 raised to an exponent (2^n). Here are a few examples:
  • 2^0 = 1
  • 2^1 = 2
  • 2^2 = 4
  • 2^3 = 8
  • 2^4 = 16
Understanding these values helps in various mathematical problems. In our exercise, knowing 8 and 16 as 2^3 and 2^4 made it easier to set and solve the inequality. It simplified our job of understanding how 2^蟺 fits within a range. Since 蟺 is a bit over 3, it makes sense that 2^蟺 is between 2^3 (8) and 2^4 (16).
The Value of Pi (蟺)
Pi (蟺) is one of the most famous constants in mathematics. It represents the ratio of a circle's circumference to its diameter and is approximately 3.14159. Let's break down why 蟺 is important:
  • Pi (蟺) is an irrational number, meaning it cannot be expressed as a simple fraction.
  • Its decimal representation never ends and does not repeat.
  • Common approximations include 3.14 or the fraction 22/7 for simpler calculations.
In our problem, knowing that 蟺 is roughly 3.14159 was key in determining where 2^蟺 lies between powers of 2. Since 蟺 is slightly greater than 3 but less than 4, we concluded that 2^蟺 is indeed greater than 8 (2^3) but less than 16 (2^4).

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Most popular questions from this chapter

As of May 2016 , the highest price paid for a painting was 300 million dollars, paid in 2015 for Willem de Kooning's "Interchange." The same painting was purchased for 20.6 million dollars in \(1989 .\) Data: wsj. com, \(2 / 25 / 16\) a) Find the exponential growth rate \(k,\) and determine the exponential growth function that can be used to estimate the painting's value \(V(t),\) in millions of dollars, \(t\) years after \(1989 .\) b) Estimate the value of the painting in 2025 c) What is the doubling time for the value of the painting? d) How many years after 1989 will it take for the value of the painting to reach 1 billion dollars?

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