Question

Two circular cylinders of equal volume have their heights in the ratio 1:2 find the ratii od their radii

Answer 1

Answer,

Let the radii of circular cylinders be

r and R

heights be h and H

and Volumes be v and V respectively.

Now given that Volumes are equal.

Therefore --

v = V

Now,

$volume \: of \: cylinder \: = \: \pi \times {r}^{2} \times h$

Therefore --

$v = \pi \times r {}^{2} \times h \\ \\ and \\ \\ V = \pi \times {R}^{2} \times H$

$= > \: \pi \times r {}^{2} \times h = \pi \times {R}^{2} \times H \\ \\ \frac{\pi \times {r}^{2} \times h}{\pi \times {R}^{2} \times H } = 1 \\ \\ cut \: \pi \: with \: \pi \\ \\ ( \frac{r}{R} ) {}^{2} \times \frac{h}{H} = 1$

$now \: \frac{h}{H} = \frac{1}{2} \\ \\ therefore \\ \\ ( \frac{r}{R} ) {}^{2} \times \frac{1}{2} = 1 \\ \\ ( \frac{r}{R} ) {}^{2} = \frac{2}{1} \\ \\ \frac{r}{R} = \sqrt{ \frac{2}{1} } \\ \\ \frac{r}{R} = \frac{ \sqrt{2} }{1}$

Therefore ratio of radii =

r : R

=> $\sqrt{2} : 1$

HOPE IT WOULD HELP YOU

Answer 2

Answer:

Solve for one height value equaling a multiple of the other, then substitute it into the formula for a cylinder's volume and use algebra to obtain

r1r2=√2.

Explanation:

We have that the formula for a cylinder's volume is

Vcylinder=πr2h

Where r is the radius and h is the height. The given height ratio is

h1h2=12

Where h1 represents the height of the first cylinder, and h2 represents that of second cylinder. We could solve for

h1=h22

or

h2=2h1

both of which we can use, if we were to consider the ratio between their volumes being equal:

π(r1)2h1π(r2)2h2=1

We could multiply the volume of the second cylinder to both sides to get

π(r1)2h1=π(r2)2h2

Now, let's see, what can we do? We can substitute h2=2h1:

π(r1)2h1=π(r2)22h1

It seems that π and h1 both cancel out:

(r1)2=2(r2)2

Let's divide by (r2)2:

(r1)2(r2)2=2

And take the square root:

r1r2=√2

We have just solved for the ratio between the radii, √2.
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Hope it will help you

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