Question

the volume of the a sphere whoes diameter is 18 cm is cubic cm . if it's diameter were reduced by half, it's volume would be of its original volume

Answer 1

The new volume is 8 times smaller than the original volume

Explanation:

we know that

If two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cube

Let

z-----> the scale factor

x ----> the volume of the reduced sphere

y ----> the volume of the original sphere

so

`z^(3)=(x)/(y)`

we have

`z=1/2` ----> scale factor

substitute

`(1/2)^(3)=(x)/(y)`

`(1/8)=(x)/(y)`

`x=(y)/(8)`

therefore

The new volume is 8 times smaller than the original volume

Verify

The volume of the original sphere is

`r=18/2=9\ cm` ---> the radius is half the diameter

`V=(4)/(3)\pi (9)^(3)=972\pi \ cm^(3)`

the volume of the reduced sphere is

`r=9/2=4.5\ cm` ---> the radius is half the diameter

`V=(4)/(3)\pi (4.5)^(3)=121.5\pi \ cm^(3)`

Divide the volumes

`972\pi \ cm^(3)/121.5\pi \ cm^(3)=8`