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Chapter 5: Problem 24

Find the geometric mean between 25 and 144.

Short Answer

Expert verified
The geometric mean is 60.

Step by step solution

01

- Understand the Geometric Mean

The geometric mean of two numbers is the square root of their product. If the two numbers are denoted as a and b, then the geometric mean (GM) is given by the formula: \[ GM = \sqrt{a \, b} \]
02

- Identify the Numbers

In this problem, the two numbers are 25 and 144.
03

- Substitute the Numbers into the Formula

Substitute 25 and 144 into the geometric mean formula: \[ GM = \sqrt{25 \, \cdot \, 144} \]
04

- Multiply the Numbers

Calculate the product of 25 and 144:\[ 25 \, \cdot \, 144 = 3600 \]
05

- Compute the Square Root

Find the square root of 3600:\[ GM = \sqrt{3600} = 60 \]

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Key Concepts

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

geometry
Geometry isn't just about shapes; it鈥檚 also about understanding different types of means, like the geometric mean. This concept is essential in both pure and applied mathematics. Here, we look at how geometric principles apply to understand the average of two numbers. When you calculate the geometric mean, you essentially preserve the multiplicative relationship between numbers. This is especially handy in maintaining proportionality in geometric figures.
mathematics problem-solving
Mathematics problem-solving often involves breaking down the problem into understandable steps. This is evident in the geometric mean calculation. By following a logical sequence of steps, one can simplify complex problems. Let's recap the steps:
  • Understand the formula: \(\text{GM} = \sqrt{a \cdot b}\)
  • Identify given numbers (25 and 144).
  • Insert these numbers into the formula: \(\text{GM} = \sqrt{25 \cdot 144}\)
  • Calculate the product \(\text{25 \cdot 144 = 3600}\)
  • Find the square root: \(\text{GM} = \sqrt{3600} = 60\)

This approach ensures a clear path to the solution, leaving room for fewer mistakes.
square root
The square root is a critical concept in understanding the geometric mean. Mathematically, the square root of a number is a value that, when multiplied by itself, gives the original number. In our example, calculating the square root of 3600 involves:
  • Knowing that \(\sqrt{x}\) represents the square root of \(x\)
  • Recognizing perfect squares, like 3600 which is \(60^2\)
  • Using calculators or methods like prime factorization if the numbers are large or unknown

Thus, the final step in finding the geometric mean involves ensuring you correctly find \(\sqrt{3600}=60\). Practice this with different sets of numbers to become proficient.

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