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The radius of the hemisphere and the height of a right circular cylinder are equal. Find out the ratio of their radii.

seo-qna
Last updated date: 20th Jun 2024
Total views: 414k
Views today: 8.14k
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 $\dfrac{a}{b}$.