The volume of a right circular cone is $150\pi \text{cm}{3}^{}$ and its height is $18\text{cm}$. Find its slant height (in cm) (correct to $2$ decimal places).

The correct option is 18.68.

To find the slant height of the right circular cone, we first need to determine the radius of its base using the given volume and height.
The volume $V$ of a right circular cone is given by the formula:
$V=\frac{1}{3}\pi r^{2}h$
where:
• $r$ is the radius of the base of the cone,
• $h$ is the height of the cone.

We are given:
• Volume, $V=150\pi {\text{cm}}^3$
• Height, $h=18\text{cm}$

Substitute these values into the volume formula:
$150\pi =\frac{1}{3}\pi r^2(18)$

Simplify the right side of the equation:
$150\pi =6\pi r^2$

Divide both sides by $6\pi$ to solve for $r^2$:
$r^2=\frac{150\pi }{6\pi }$
$r^2=25$
Taking the square root, we get:
$r=5\text{cm}$

The relationship between the radius ($r$), height ($h$), and slant height ($l$) of a right circular cone is given by the Pythagorean theorem:
$l^2=r^2+h^2$

Substitute the values of $r^2$ and $h$:
$l^2=25+{18}^2$
$l^2=25+324$
$l^2=349$

Now, find the slant height $l$ by taking the square root:
$l=\sqrt{349}\approx 18.6815\text{cm}$

Rounding to 2 decimal places, the slant height is 18.68 cm.