Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Find the geometric mean of 20 and 25 .

Step-by-Step Solution

Answer

The geometric mean of 20 and 25 is approximately 22.36.

Step 1: Understand the Formula for Geometric Mean

The geometric mean of two numbers a and b is calculated using the formula \( \sqrt{a \times b} \). For this problem, a = 20 and b = 25.

Step 2: Set Up the Expression

Using the numbers provided, substitute into the formula: \( \sqrt{20 \times 25} \). You are finding the square root of the product of the two numbers.

Step 3: Calculate the Product

Multiply 20 by 25 to get the product: \( 20 \times 25 = 500 \).

Step 4: Find the Square Root of the Product

Next, calculate \( \sqrt{500} \). This requires finding a number that when squared equals 500.

Step 5: Simplify the Square Root

\( \sqrt{500} \) can be simplified by recognizing it as \( \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5} \).

Step 6: Approximate the Square Root

To approximate \( 10\sqrt{5} \), you need the approximate value of \( \sqrt{5} \), which is about 2.236. Multiply: \( 10 \times 2.236 = 22.36 \).

Key Concepts

Square Root

In mathematics, the square root of a number is a value that, when multiplied by itself, gives that original number.
For example, the square root of 9 is 3 because 3 times 3 equals 9.
When calculating the geometric mean, you often encounter the square root operation as part of the formula.

• The symbol for square root is \( \sqrt{} \).
• It is essential to understand that the square root of a product is equal to the product of the square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).

Finding the square root of a number that is not a perfect square often results in a decimal.
In such cases, simplifying the square root can help make calculations more manageable, especially when approximation is involved.

Product Calculation

Product calculation is the process of multiplying two or more numbers together. For the problem at hand, you need to calculate the product of 20 and 25.
Simply put:

• Multiply the two numbers: 20 and 25.
• The result is 500, which is the product used in further calculations.

This step is crucial as it prepares the base from which you find the geometric mean.
The product forms the radicand in the square root, representing the measure over which you'll establish your mean.
Remember, when multiplying numbers, the order of multiplication does not matter, as it is commutative.

Simplification

Simplification involves breaking down complex expressions into simpler components.
In the calculation of the geometric mean, you'll deal with simplifying the square root.
Take for instance \( \sqrt{500} \). This isn't a simple square root and can be decomposed into:

• Prime factor 500 as \( 100 \times 5 \).
• \( \sqrt{500} = \sqrt{100 \times 5} \).
• This further simplifies to \( \sqrt{100} \times \sqrt{5} \).
• Since \( \sqrt{100} = 10 \), you end up with \( 10\sqrt{5} \).

Simplification is useful for understanding deeper relationships within mathematical expressions and makes calculation of approximations easier.
It will also help ensure calculations are clear and correct, especially when dealing with irrational numbers.

Approximation

Approximation is the process of finding a value that is close enough to the exact number for practical purposes.
This becomes necessary when dealing with irrational numbers that cannot be neatly expressed as a fraction or finite decimal.
For example, \( \sqrt{5} \) does not result in a whole number, but you can approximate it:

• \( \sqrt{5} \approx 2.236 \).

Multiplying this result by 10, as derived from our earlier simplification, you get the approximated geometric mean, \( 22.36 \).
Approximations are important in most calculations to provide a meaningful and usable result, especially in real-world applications where exact numbers can be cumbersome.