Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Find the geometric mean of 12 and 48 .

Step-by-Step Solution

Answer

The geometric mean of 12 and 48 is 24.

Step 1: Understand the Geometric Mean

The geometric mean of two numbers, say \(a\) and \(b\), is the square root of their product. It can be represented by the formula \( \sqrt{a \times b} \). This measure is useful for finding the central tendency of products or exponential rates.

Step 2: Compute the Product

Multiply the given numbers 12 and 48. Thus, compute \(12 \times 48\). The product is 576.

Step 3: Calculate the Square Root

Take the square root of the product calculated in Step 2. Compute \(\sqrt{576}\). The result is 24.

Key Concepts

Central Tendency

Central tendency refers to the middle point around which data values tend to cluster. Imagine marks in a class test; most students will score around a central score, indicating the general performance level. In statistics, central tendency can be represented by different measures such as the mean, median, and mode.

While the arithmetic mean is often the most familiar, for data involving rates or percentages, the geometric mean is a better representation. This is because it doesn't get disproportionately affected by extremely high or low values. Unlike the arithmetic mean, which adds the numbers up and divides by the count, the geometric mean multiplies the numbers and then takes a root (e.g., square root for two numbers).

Understanding where different types of means apply is crucial. For example, use the geometric mean to determine the average growth rates over time for financial or population studies. This is because it accounts for compounding, offering a clearer picture of average performance across multiple periods.

Exponential Rates

Exponential rates describe processes that grow or decay at a rate proportional to their current value. Think of how money grows in a bank account with compound interest or bacteria population growing by doubling.

The geometric mean is essential in calculating these rates because it facilitates understanding the average rate of return for investments. For example, if you have consecutive returns over several periods, the geometric mean provides the constant rate that would yield the same final amount as the actual sequence of varied returns.

In essence, the geometric mean smoothens out fluctuations over consecutive periods, offering a more consistent measure compared to arithmetic means, which may not account for the compounding nature of growth or decay. This makes it invaluable in fields where exponential progression is common, such as finance, epidemiology, and environmental science.

Square Root

The square root is a mathematical function that helps us find a number which, when multiplied by itself, gives the original number. In the context of the geometric mean, it plays a pivotal role in determining the central tendency of numbers.

To calculate the geometric mean, we use the square root because it essentially unravels the product of the numbers, revealing the balance point. For two numbers, it's simple; you multiply them and then take the square root, like the geometric mean of 12 and 48 where \[\sqrt{12 \times 48} = \sqrt{576} = 24\].

Understanding how to compute square roots is crucial, not only for solving geometric mean problems but also as a foundational skill in algebra and higher mathematics. Square roots help in geometry to solve areas and volumes, and in physics for wave calculations, making them widely applicable.