In a solid cylindrical vessel, the numerical value of its volume is equal to four times the numerical value of its curved surface area. Which of the following conditions must hold true regarding its dimensions?

The correct option is r = 8.

Step-by-step Explanation:

Let $r$ be the radius of the solid cylindrical vessel and $h$ be its height.

The formula for the volume ($V$) of a cylinder is given by:

$V=\pi r^{2}h$

The formula for the curved surface area ($CSA$) of a cylinder is given by:

$CSA=2\pi rh$

According to the given condition, the numerical value of the volume is equal to four times the numerical value of its curved surface area:

$V=4\times CSA$

Substitute the formulas for volume and curved surface area into the equation:

$\pi r^{2}h=4\times \left(2\pi rh\right)$

Simplify the right side of the equation:

$\pi r^{2}h=8\pi rh$

Since the radius $r$ and height $h$ are physical dimensions of a solid vessel, they must be non-zero positive numbers ($r>0$ and $h>0$). Therefore, we can divide both sides of the equation by $\pi rh$:

$\frac{\pi r^{2}h}{\pi rh}=\frac{8\pi rh}{\pi rh}$

This simplifies directly to:

$r=8$

Thus, the condition that must hold true regarding the dimensions of the vessel is $r=8$.