<s>[INST] Volume of a cuboid whose diagonal and surface area is known The sum of length, breadth and depth of cuboid is $19$cm and its diagonal is $5\sqrt{5}$cm.
Its volume is:

a) 125
b) 236
c) 361
d) 486

Solution:
$$\ell^2 + b^2 +h^2 = 125\quad\text{ and }\quad \ell+b+h=19,$$
How can i find volume from this information?
 [/INST] $l+b+h=19$
$l^2+b^2+h^2=125$
From these 2 equations, we get
$2(lb+bh+hl) = (l+b+h)^2-(l^2+b^2+h^2)
= 19^2-125 = 236 = S$
This is coincidentally the Surface area of the cuboid.
The Volume $lbh = \frac{S}{2(l^{-1}+b^{-1}+h^{-1})}$.
Assuming $l=b$, we can find $l$ and $h$ to be $\frac{19-\sqrt{7}}{3}$ and $\frac{19+2\sqrt{7}}{3}$ and the Volume is then $240.63$.
Max Volume is when $l=b=h$ and we get $254.037$, so answers $c$ and $d$ are our. All we can say is that answer $b$ ($236$) is possibly right as Volume. I don't think you can actually find the Volume. 
</s>