Answer:
The volume is increased in a ratio of 64/27
Explanation:
The Volume of a Sphere
The volume of a sphere of radius r is given by:
[tex]\displaystyle V=\frac{4}{3}\cdot \pi\cdot r^3[/tex]
If the radius was changed to r', the new volume would be:
[tex]\displaystyle V'=\frac{4}{3}\cdot \pi\cdot r'^3[/tex]
Dividing the latter equation by the first one:
[tex]\displaystyle \frac{V'}{V}=\frac{\frac{4}{3}\cdot \pi\cdot r'^3}{\frac{4}{3}\cdot \pi\cdot r^3}[/tex]
Simplifying:
[tex]\displaystyle \frac{V'}{V}=\frac{r'^3}{r^3}[/tex]
Or, equivalently:
[tex]\displaystyle \frac{V'}{V}=\left(\frac{r'}{r}\right)^3[/tex]
Since r':r = 4:3, thus:
[tex]\displaystyle \frac{V'}{V}=\left(\frac{4}{3}\right)^3[/tex]
[tex]\displaystyle \frac{V'}{V}=\frac{4^3}{3^3}[/tex]
[tex]\displaystyle \frac{V'}{V}=\frac{64}{27}[/tex]
The volume is increased in a ratio of 64/27
