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

Question

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

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    0
    2022-01-04T15:55:45+00:00

    Answer:

    The volume is changed by the factor of 1/8

    Step-by-step 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

    0
    2022-01-04T15:55:53+00:00

    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=\frac{r}{2}\\\\ h=\frac{h}{2} \end{cases}\implies V=\pi \left( \frac{r}{2} \right)^2\left( \frac{h}{2} \right)\implies V=\pi \left(\frac{r^2}{2^2}  \right)\frac{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.

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