Consider a cuboid of sides 2 x , 4 x and 5 x and a closed hemisphere of radius r . If the sum of their…

Let sum of surface area

 $k=3\pi r^2+76x^2$

$\because k$ is constant so $\frac{dk}{dx}=0$

$\Rightarrow 6\pi r\frac{dr}{dx}+2\times 76x=0$

$\Rightarrow \frac{dr}{dx}=-\frac{76x}{3\pi r} \cdots \left(1\right)$

Now total volume $V=\frac{2}{3}\pi r^3+40x^3$

For maximum volume $\frac{dV}{dx}=0$

$\Rightarrow 2\pi r^2\frac{dr}{dx}+120x^2=0 \cdots \left(2\right)$

$\Rightarrow 2\pi r^2\left(-\frac{76x}{3\pi r}\right)+120x^2=0$

$\Rightarrow 8x\left[-\frac{19r}{3}+15x\right]=0$

$\Rightarrow \frac{x}{r}=\frac{19}{45}$