Question

If root p - root q = 20 then what is the maximum value of (p - 5q)/100

Answer 1

Given:

$\bold{\sqrt{p} -\sqrt{q} =20}$

To prove:

Find maximum value of =  $\frac{p-5q}{100}$

Solution:

formula:

$(a+b)^{2}= a^2+b^2+2ab$

let:

$\Rightarrow \sqrt{p} -\sqrt{q} =20$

$\Rightarrow$$\sqrt{p} = \sqrt{q} +20 .....(i)$

square the above equation:

$p= q+400+40\sqrt{q}$

put the above value in to equation:

$\bold {\ Equation:} \\\\ \Rightarrow \frac{p-5q}{100} \\\\ \Rightarrow \frac{q+400+40\sqrt{q} -5q}{100} \\\\ \Rightarrow \frac{400+40\sqrt{q} -4q}{100}$

Let the above equation as a method:

$\Rightarrow f(x) = \frac{400+40\sqrt{q} -4q}{100} \\\\\ Calculating \ derivative \\\\\Rightarrow f'(x) \ = \frac{20 }{\sqrt{q}} -4} \\\\\Rightarrow f'(x) \ = \sqrt{q} = 5$

if square the above value q= 25, and put the value of q and √q in method f(x):

 $\Rightarrow f(x) = \frac{400+40\times 5 -4 \times 25}{100} \\\\\Rightarrow f(x) = \frac{400+200-100}{100} \\\\\Rightarrow f(x) = \frac{500}{100} \\\\\Rightarrow f(x) = 5$

The maximum value of (p-5q)/100 is = 5