Answer

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**Hint:**Here take the radius of hemisphere and the height of a right circular cylinder equal as given in the question. Now use the volume of hemisphere and right circular cylinder $\dfrac{{2\pi {r^3}}}{3}$ and $\pi {R^2}h$ respectively. So, by taking the ratio of volumes we will obtain the ratios of their radii.

**Complete step-by-step answer:**

Let radius of hemisphere be r,

Let height of a right circular cylinder be h and

Let the radius of a right circular cylinder be R.

Now, $r = h$, as it is given that the radius of the hemisphere and the height of a right circular cylinder are equal.

Now we know that,

Volume of hemisphere is $\dfrac{{2\pi {r^3}}}{3}$ -(1)

Volume of right circular cylinder is $\pi {R^2}h$ -(2)

Now we will consider the ratio of volume of hemisphere and cylinder,

\[

volume{\text{ of hemisphere : volume of right circular cylinder}} \\

\dfrac{{2\pi {r^3}}}{3}:\pi {R^2}h \\

{\text{2}}{{\text{r}}^3}:3{R^2}h \\

{\text{ }} \\

\]

Now using $r = h$,

\[{\text{2}}{{\text{r}}^2}:3{R^2}\]

$

\sqrt 2 r:\sqrt 3 R \\

\dfrac{r}{{\sqrt 3 }}:\dfrac{R}{{\sqrt 2 }} \\

\dfrac{r}{R} = \dfrac{{\sqrt 3 }}{{\sqrt 2 }} \\

\dfrac{r}{R} = \sqrt {1.5} \\

\dfrac{r}{R} = 1.2247 \\

$

So, the ratio of the radii of the hemisphere and the right circular cylinder is $1.2247$ approximately.

**Note:**We should note that if

$

\dfrac{x}{a}:\dfrac{y}{b} \\

\\

$

Then we say that

$\dfrac{x}{y} = \dfrac{a}{b}$

Which means the ratio of x and y is $\dfrac{a}{b}$.

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