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Mean proportional of two number is given in the question but numbers are not

So first we have to let a, b are the required numbers.

Formula of mean proportional

$\sqrt {ab} = $mean proportional

$\sqrt {ab} = 28$

Now take the square both side

${(\sqrt {ab} )^2} = {28^2}$

$ab = 28.28$

$ab = 784$

So we can find the value of number a

$a = \dfrac{{784}}{b}$ ……… equation (1)

We have the third proportional given in the question that is 224

The formula of third proportional

$ac = {b^2}$

$c = \dfrac{{{b^2}}}{a}$

Put the values

Here c is the third proportional

$224 = \dfrac{{{b^2}}}{a}$

Now put the value of a

$224 = \dfrac{{{b^2}}}{{\dfrac{{784}}{b}}}$

Simplifying the equation

$224 = {b^2}.\dfrac{b}{{784}}$

Multiply the R.H.S

$224 = \dfrac{{{b^3}}}{{784}}$

Apply the cross-multiplication method

${b^3} = 224.784$

${b^3} = $175616

$b{ = ^3}\sqrt {175616} $

$b = 56$

So here we the second number

We can find the first number a with the help of equation (1)

$a = \dfrac{{784}}{b}$

$a = \dfrac{{784}}{{56}}$

$a = 14$

Hence, we have both the numbers

First is 14 and the second number is 56.

Mean proportional =$\sqrt {a.b} $

We have a = 14

And b = 56

After putting the values, we get

= $\sqrt {14.56} $

$ = \sqrt {784} $

$ = 28$

So here we get mean proportional that is already given in the question

Our answer is correct by alternative checking method.