Question

If the height of a cylinder is doubled, by what number must the radius of the base be multiplied so that the resulting cylinder has the same volumes as the original cylinder?
A. 4
B. 1/√2
C. 2
D.1/2

Answer 1

Let volume of original cylinder be V cm³

Let height be h

Let radius be r

V = πr²h

Height cylinder is doubled.

So, V' = 2πr²h

V' = 2V

So to make V' = V

Radius should be multiplied by 1/√2

Now V' = 2π(r/√2)²h

V' = 2πr²h/2

V' = πr²h

V' = V

So option B is correct.

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Answer 2

Answer:

The radius of the base multiplied by $\dfrac{1}{\sqrt{2} }$  so that resultant cylinder has same volume as original cylinder  

Step-by-step explanation:

Given as :

For original cylinder

The height of cylinder = h  unit

The radius of cylinder = r  unit

So, Volume of cylinder = $V_1$ = π × r² × h

where r = radius

          h = height

Again

For Resulting cylinder

The height of cylinder = H = 2 h

The radius of cylinder = R = x r

So, Volume of cylinder = $V_2$ = π × R² × H

i.e   $V_2$ = π × (x r)² × 2 h

where r = radius

          h = height

According to question

The resultant cylinder has same volume as original cylinder

i.e     $V_1$   =   $V_2$

Or,    π × r² × h   =   π × (x r)² × 2 h

Or,   π × r² × h   =   π × x² × r² × 2 h

Or,   π × r² × h   =    π × r² × h  × x² × 2

Or, 2 x²   =  $\dfrac{\pi r^{2} h{^{2} } }{\pi r^{2} h{^{2} } }$

i.e   2 x²   = 1

Or,  x²   = $\dfrac{1}{2}$

∴    x = $\dfrac{1}{\sqrt{2} }$

Hence, The radius of the base multiplied by $\dfrac{1}{\sqrt{2} }$  so that resultant cylinder has same volume as original cylinder  . Answer