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

Answer:

The new volume is 8 times smaller than the original volume

Step-by-step 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

$z^{3}=\frac{x}{y}$

we have

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

substitute

$(1/2)^{3}=\frac{x}{y}$

$(1/8)=\frac{x}{y}$

$x=\frac{y}{8}$

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

$V=\frac{4}{3}\pi (9)^{3}=972\pi \ cm^{3}$

the volume of the reduced sphere is

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

$V=\frac{4}{3}\pi (4.5)^{3}=121.5\pi \ cm^{3}$

Divide the volumes

$972\pi \ cm^{3}/121.5\pi \ cm^{3}=8$