If x is the mean proportional between 12.8 and 64.8, then the value of x is:
28.8
Understanding Mean Proportional Concepts
The question asks us to determine the value of x, which is defined as the mean proportional between the numbers 12.8 and 64.8. The concept of mean proportional is fundamental in understanding ratios and proportions.
If a number 'x' is the mean proportional between two numbers 'a' and 'b', it means that the ratio of the first number to the mean proportional is equal to the ratio of the mean proportional to the second number. This can be expressed mathematically as:
$$ a : x :: x : b $$
This proportion can be written in fractional form:
$$ \frac{a}{x} = \frac{x}{b} $$
By cross-multiplying the terms, we get the relationship:
$$ x^2 = a \times b $$
Therefore, to find the mean proportional 'x', we need to calculate the square root of the product of the two numbers 'a' and 'b':
$$ x = \sqrt{a \times b} $$
Calculating the Value of x
We are given the two numbers:
- a = 12.8
- b = 64.8
We need to find the value of x, the mean proportional between these numbers.
Step 1: Compute the product of the numbers
First, we multiply the two given numbers together:
$$ \text{Product} = 12.8 \times 64.8 $$
Performing the multiplication:
$$ 12.8 \times 64.8 = 829.44 $$
Step 2: Calculate the square root of the product
Next, we find the square root of the result obtained in Step 1:
$$ x = \sqrt{829.44} $$
To find the square root of 829.44, we can estimate or use a calculator. We are looking for a number that, when multiplied by itself, equals 829.44.
Let's try squaring numbers around the estimated value. Since $20^2 = 400$ and $30^2 = 900$, the value should be between 20 and 30. Let's check 28.8:
$$ (28.8)^2 = 28.8 \times 28.8 = 829.44 $$
So, the square root of 829.44 is exactly 28.8.
The value of x, the mean proportional between 12.8 and 64.8, is 28.8.
Matching the Result with Options
The calculated value of x is 28.8. Now, let's compare this value with the options provided:
- Option 1: 32
- Option 2: 34
- Option 3: 28.8
- Option 4: 26.4
Our calculated result, 28.8, matches exactly with Option 3.
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