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Chapter 8: Problem 48

Find the geometric mean between each pair of numbers. 6 and 9

Short Answer

Expert verified
The geometric mean is approximately 7.35.

Step by step solution

01

Understand the Geometric Mean

The geometric mean of two numbers is the square root of their product. It's useful for finding an average value that represents proportional growth, commonly used in fields like finance and science.
02

Calculate the Product

Multiply the given numbers, 6 and 9, together: \[ 6 \times 9 = 54 \]
03

Find the Square Root

Calculate the square root of the product obtained in Step 2: \[ \sqrt{54} \] This simplifies to approximately \( 7.35 \), considering this value to two decimal places.

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Key Concepts

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

Product of Numbers
The first step in finding the geometric mean between two numbers is to calculate the product of those numbers. This simply means multiplying them together. In our example with the numbers 6 and 9, the calculation is straightforward:
  • Start with the numbers 6 and 9.
  • Multiply them to find their product: \(6 \times 9 = 54\).
This product, 54, serves as the foundation for the next step when we calculate the geometric mean. Calculating the product is an essential step because it combines the two values in a way that reflects their joint contribution as a single figure.
Square Root Calculation
After obtaining the product of the numbers, the next step is to find the square root. The square root helps in balancing out the product to find a mean value. To find the geometric mean of 6 and 9, you take the square root of their product, which is 54. This is calculated as follows:
  • The operation is: \(\sqrt{54}\).
  • When computed, the square root of 54 is approximately 7.35.
Finding the square root ensures that we bring the multiplied figures back down into a more manageable range, anchoring them around a common mean.
Proportional Growth
The concept of proportional growth is at the heart of why geometric mean is significant, especially in fields like finance and science. The geometric mean gives an average that accounts for exponential growth rates or proportional changes:
  • Unlike the arithmetic mean, the geometric mean emphasizes the effect of compounded rates.
  • This makes it particularly useful for understanding growth patterns, such as interest rates or population growth, where the growth rate is proportional from one period to the next.
By using the geometric mean, we ensure that the calculated mean reflects realistic scenarios of growth, rather than simple addition of values.
Average Value
The geometric mean gives us an "average" that is more appropriate in certain contexts than the arithmetic mean. Here's why it might be considered a true "average" in some scenarios:
  • In situations that involve rates of return or varying productivities, a simple arithmetic mean might not be accurate as it does not account for the compound nature of these situations.
  • With the geometric mean of 6 and 9 being approximately 7.35, we have an average that is weighted for multiplicative scenarios.
This type of average ensures a balanced approach in understanding data sets that exhibit non-linear growth, providing insights that regular mean values might miss.

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Most popular questions from this chapter

Determine whether each set of measures contains the sides of a right triangle. Then state whether they form a Pythagorean triple. $$20,21,31$$

Find two pairs of numbers with a geometric mean of 12 .

Simplify each expression by rationalizing the denominator. $$\frac{24}{\sqrt{2}}$$

Find the measure of each angle to the nearest tenth of a degree. $$\sin B=0.5127$$

Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple. $$4,5,6$$
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