Question

What is the mean proportional between (15 + 200) and (27 – V648)
a) 4
b) 14v7)
c) 3V5
d) 5 V3​

Answer 1

Answer:

3v5

STEP

Here,a=15+200−−−√,b=27-648−−−√

mean proportional between two numbers

=ab−−√

mean proportional between (15+200−−−√) and (27-648−−−√

=(15+200−−−√)−−−−−−−−−−√(27−648−−−√)

=(15×27)−15×648−−−√+27200−−−√−(200−−−√×648−−−√)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√

=405−(15×182–√)+(27×102–√−102–√×182–√)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√

=405−2702–√+2702–√−180×2−−−−−−−−−−−−−−−−−−−−−−−−−√

=405−180×2−−−−−−−−−−−√

=405−360−−−−−−−−√ =45−−√ =35–√

Answer 2

The mean proportional between $15 + \sqrt{200} \text{ and } 27 - \sqrt{648} \text{ is } 3 \sqrt{5}$

Solution:

Given are:

$15 + \sqrt{200} \text{ and } 27 - \sqrt{648}$

mean proportional between two numbers = $\sqrt{ab}$

Therefore,

$\text{mean proportional } = \sqrt{(15 + \sqrt{200}) \times (27 - \sqrt{648}) } \\\\= \sqrt{(15 \times 27) -15 \times \sqrt{648} + 27 \sqrt{200} -(\sqrt{200} \times \sqrt{648})}\\\\\\ = \sqrt{405 - (15 \times 18 \sqrt{2})+ (27 \times 10 \sqrt{2} - 10 \sqrt{2} \times 18 \sqrt{2})}\\\\= \sqrt{ 405 - 270 \sqrt{2} + 270 \sqrt{2} -180 \times 2}\\\\= \sqrt{ 405 - 360}\\\\= \sqrt{45}\\\\= 3 \sqrt{5}$

Thus mean proportional between $15 + \sqrt{200} \text{ and } 27 - \sqrt{648} \text{ is } 3 \sqrt{5}$

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