Question

two right circular cylinder of equal volume have their radii in the ratio square root 2:1 find the ratio of their heights
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Answer 1

Answer :-

$\\\;\large{\underbrace{\underline{\textsf{Question's Analysis\;:-}}}}$

The concept of Volume of Cylinders has been applied in this case. Two right circular cylinders of identical volume with radii that are in proportion are given. The ratio of their heights is what we're supposed to figure out. We can apply the formula of Volume of Cylinder. From this formula, we can find the ratio of the height of the right circular cylinder.

Let's solve the problem !!

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Equations used :-

$\\\;\boxed{\sf{\;Volume_{(Cylinder\;1)}=Volume_{(Cylinder\;2)}\;}}$

$\\\;\boxed{\sf{\;Volume_{(Cylinder)}=\bf{\pi{r}^{2}h\;}}}$

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Solution :-

Let us assume that, the radius of the base of two right circular cylinders be R & r respectively.

Let us assume that, the height of the base of two right circular cylinders be H & h respectively.

Given that,

» Radius of the first cylinder = R

» Radius of the second cylinder = r

Then, the ratio of radii of two right circular cylinders = R : r = √2 : √1.

Or we can say that, the ratio of radii of two right circular cylinders = R/R = √2/1.

As it is given, two right circular cylinders have equal volume. Therefore our equation becomes,

$\\\;\implies \bf{\;Volume_{(Cylinder\;1)}=Volume_{(Cylinder\;2)}}$

Now, by using the above equation, we obtain:

$\\\;\implies \sf{\;\pi{R}^{2}H = \pi{r}^{2}h}$

$\\\;\implies \sf{\;{R}^{2}H = {r}^{2}h}$

$\\\;\implies \sf{\dfrac{{R}^{2}}{{r}^{2}} = \dfrac{H}{h}}$

$\\\;\implies \sf{\left(\dfrac{R}{r}\right)^{2} = \dfrac{H}{h}}$

Now on substituting the known value in the above equation, we get:

$\\\;\implies \sf{\left(\sqrt{\dfrac{2}{1}}\right)^{2} = \dfrac{H}{h}}$

$\\\;\implies \sf{\dfrac{2}{1} = \dfrac{H}{h}}$

$\\\;\implies \sf{2:1 = H:h}$

$\\\;\implies \bf{H:h = 2:1}$

Thus, the ratio of their heights is 2:1.

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More formulas to know :-

• Volume of cylinder = πr²h

• T.S.A of cylinder = 2πrh + 2πr²

• Volume of cuboid = l × b × h

• C.S.A of cuboid = 2(l + b)h

• T.S.A of cuboid = 2(lb + bh + lh)