Calculating the Third Proportional and Finding a Value

The question asks us to find the positive value of \(y\), given that the third proportional of \(3x^2\) and \(4xy\) is 48.

Let's first understand what the third proportional is. If we have two numbers, say \(a\) and \(b\), their third proportional, say \(c\), is a number such that \(a, b, c\) are in continuous proportion. This means the ratio of the first to the second is equal to the ratio of the second to the third. Mathematically, this is expressed as:

\(\frac{a}{b} = \frac{b}{c}\)

In this problem, the first number is \(a = 3x^2\), the second number is \(b = 4xy\), and the third proportional is \(c = 48\). We can set up the proportion using the formula:

\(\frac{3x^2}{4xy} = \frac{4xy}{48}\)

Now, we need to solve this equation for \(y\). We can cross-multiply to eliminate the denominators:

\((3x^2) \times 48 = (4xy) \times (4xy)\)

Let's simplify both sides of the equation:

• Left side: \(3x^2 \times 48 = 144x^2\)
• Right side: \(4xy \times 4xy = 16x^2y^2\)

So the equation becomes:

\(144x^2 = 16x^2y^2\)

Assuming \(x \neq 0\) (as if \(x=0\), both original numbers are 0, and the concept of a non-zero third proportional doesn't typically apply), we can divide both sides of the equation by \(16x^2\) to isolate \(y^2\):

\(\frac{144x^2}{16x^2} = y^2\)

Simplify the left side:

\(\frac{144}{16} = y^2\)

\(9 = y^2\)

To find the value of \(y\), we take the square root of both sides:

\(y = \pm\sqrt{9}\)

\(y = \pm 3\)

The possible values for \(y\) are 3 and -3. The question specifically asks for the positive value of \(y\).

Therefore, the positive value of \(y\) is 3.

Revision Table: Third Proportional Concepts

Additional Information on Proportions

Besides the third proportional, there are other related concepts in proportions:

• Mean Proportional: For two numbers \(a\) and \(c\), the mean proportional (or geometric mean) is the number \(b\) such that \(a, b, c\) are in continuous proportion (\(\frac{a}{b} = \frac{b}{c}\)). This simplifies to \(b^2 = ac\), so \(b = \pm\sqrt{ac}\).
• Fourth Proportional: For three numbers \(a, b, c\), the fourth proportional is the number \(d\) such that \(a, b, c, d\) are in proportion (\(\frac{a}{b} = \frac{c}{d}\)). This can be solved for \(d\) as \(d = \frac{bc}{a}\).

Understanding these different types of proportionals is important for solving various problems involving ratios and proportions.