Question

If the height and diameter of the cylinder are halved, by what factor will the volume of the cylinder change?

Answer 1

volume of a cylinder is V = πr²h, where r = radius and h = height.

now, if you cut the diameter by half, you also cut the radius by half, so we'd end up with r/2 instead.

if you cut the height in half, we'd end up with h/2.

then,


`\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h\quad \begin{cases} r=(r)/(2)\\\\ h=(h)/(2) \end{cases}\implies V=\pi \left( (r)/(2) \right)^2\left( (h)/(2) \right)\implies V=\pi \left((r^2)/(2^2) \right)(h)/(2) \\\\\\ V=\pi \cdot \cfrac{r^2}{4}\cdot \cfrac{1}{2}\cdot h\implies V=\cfrac{1}{4}\cdot \cfrac{1}{2}\cdot \pi r^2 h\implies V=\cfrac{1}{8}(\pi r^2 h)`

notice, the new size is just 1/8 of the original size.

Answer 2

The volume is changed by the factor of 1/8

Explanation:

The problem bothers on the volume of a cylinder

Step one

The expression for the volume of a cylinder is

V=pi*r²*h

Where r= radius of the cylinder

h= height t of the cylinder

Step two

Now we are told that the diameter and the height were halved

I.e diameter =d/2

Height =h/2

But r= (d/2)

Hence if diameter is halved radius is also halved

raduis =r/2

Also the height =h/2

Step three

Hence the factor by which the volume changes can be gotten by putting this parameter in the volume of the cylinder

V=pi*(r/2)²*h/2

V=pi*(r²/4)*h/2

V=(pi*r²h)/8

From the emerging equation the volume is changed by the factor 1/8