The mean proportional between two numbers is 28 and third proportional to them is 224 find the numbers

To find the two numbers given that their mean proportional is 28 and the third proportional to them is 224, we can proceed with the following steps:

### Step 1: Understanding Mean Proportional
The mean proportional between two numbers \(a\) and \(b\) is given by:
\[
\sqrt{a \times b} = 28
\]
This implies:
\[
a \times b = 28^2 = 784
\]

### Step 2: Understanding Third Proportional
The third proportional to \(a\) and \(b\) is a number \(c\) such that:
\[
\frac{a}{b} = \frac{b}{c}
\]
This implies:
\[
c = \frac{b^2}{a}
\]
We are given that \(c = 224\), so:
\[
\frac{b^2}{a} = 224 \quad \Rightarrow \quad b^2 = 224a
\]

### Step 3: Solve the System of Equations
Now, we have two equations:
1. \(a \times b = 784\)
2. \(b^2 = 224a\)

#### From the first equation:
\[
b = \frac{784}{a}
\]

#### Substitute \(b = \frac{784}{a}\) into the second equation:
\[
\left(\frac{784}{a}\right)^2 = 224a
\]
\[
\frac{614656}{a^2} = 224a
\]
\[
614656 = 224a^3
\]
\[
a^3 = \frac{614656}{224} = 2744
\]
\[
a = \sqrt[3]{2744} = 14
\]

#### Now, find \(b\):
\[
b = \frac{784}{14} = 56
\]

### Final Answer
The two numbers are **14** and **56**.

Would you like any additional details, or do you have any questions?

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**Related Questions:**
1. What is the general formula for finding the mean proportional?
2. How is the third proportional defined and applied in different contexts?
3. What is the significance of cube roots in mathematical problems like this?
4. How would the problem change if the third proportional were a different number?
5. Can you find the fourth proportional if given three numbers?

**Tip:** Always double-check your substitutions when solving systems of equations to avoid calculation errors.

Explore how to solve problems involving mean and third proportionals in mathematics. This detailed solution covers finding two numbers given their mean proportional and third proportional values. Suitable for advanced high school students and mathematics enthusiasts.

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