Problem 17 - Technical Mathematics with Calculus

Question

Find the mean proportional between the following. 5 and 45

Step-by-Step Solution

Answer

The mean proportional between 5 and 45 is 15.

Step 1: Understand the Concept of Mean Proportional

The mean proportional between two numbers, a and b, is the square root of the product of the two numbers. The formula to find the mean proportional (x) is given by: \(x = \sqrt{a \cdot b}\).

Step 2: Insert the Given Numbers into the Formula

Substitute the given numbers into the formula to find the mean proportional: \(x = \sqrt{5 \cdot 45}\).

Step 3: Calculate the Product Inside the Square Root

Multiply the two numbers inside the square root to get the value that we will take the square root of: \(5 \cdot 45 = 225\).

Step 4: Compute the Square Root

Find the square root of the product calculated in the previous step: \(\sqrt{225} = 15\). Therefore, the mean proportional between 5 and 45 is 15.

Key Concepts

Square Root

The concept of the square root is fundamental to various fields in mathematics, including algebra, geometry, and calculus. It is defined as a value that, when multiplied by itself, gives the original number. For example, if you are searching for the square root of a number like 25, you are looking for a number that, when multiplied by itself (\times itself), results in 25. In this case, that number is 5, since 5 \times 5 = 25.

The square root is represented by the symbol \(\sqrt{}\) and can be computed using a calculator, by hand using the long division method, or approximated using various techniques. Square roots are essential in solving quadratic equations, understanding the Pythagorean theorem in geometry, and finding mean proportional values, which include the geometric mean between two numbers.

Geometric Mean

In contrast with the arithmetic mean, which you may be familiar with when calculating the average of a set of numbers, the geometric mean is a level of central tendency that is especially useful in situations with proportional growth or products of numbers. The geometric mean between two numbers, a and b, is the square root of their product. This concept can be represented mathematically as \(\text{{geometric mean}} = \sqrt{a \times b}\).

The geometric mean is widely used beyond school mathematics. In finance, it's used to calculate average rates of return over time, considering compounding effects. In science, it helps to determine various rates such as growth rates of populations or chemicals. Understanding how to compute the geometric mean not only strengthens problem-solving skills but also opens up to practical applications in real-life scenarios.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that has wide applications in academic contexts and real-world scenarios. The process generally involves understanding the problem, devising a plan, carrying out the plan, and then looking back to check and interpret the results. For the problem of finding the mean proportional, the first step is to understand the problem, which involves recognizing that the mean proportional is the geometric mean between two given numbers.

Next, you devise a plan by determining which mathematical formula or concept applies – in this case, the formula for the geometric mean. Carrying out the plan involves applying the formula correctly, as shown by inserting the given numbers and performing calculations. The final step is to check the work to ensure the result is sensible, and in a broader context, to understand what this result might tell us about the relationship between the numbers or the situation they represent.