The volume of the solid object is 12 cubic units
Note that dilating an object with a scale factor of k will reduced or enlarge the measurement of it's side.
For the volume, we have three dimensions since its in the cubic form..
For example :
length x width x height (Three dimensions)
So to get the new volume :
[tex]V_{\text{new}}=\lbrack(V^{}_{\text{orig}})^{\frac{1}{3}}\times k\rbrack^3[/tex]The original volume must be raised to 1/3 to get one unit dimension..
[tex](units^3)^{\frac{1}{3}}=unit[/tex]Substitute the original volume and the values of k to the formula :
For k = 1/4
[tex]V_{new}=(12^{\frac{1}{3}}\times\frac{1}{4})^3_{}=\frac{3}{16}=0.188[/tex]For k = 0.4
[tex]V_{new}=(12^{\frac{1}{3}}\times0.4)^3_{}=\frac{9}{125}=0.072[/tex]For k = 1
Volume will still be the same since the scale factor is 1.
For k = 1.2
[tex]V_{new}=(12^{\frac{1}{3}}\times1.2)^3_{}=\frac{2592}{125}=20.736[/tex]For k = 5/3
[tex]V_{new}=(12^{\frac{1}{3}}\times\frac{5}{3})^3_{}=\frac{500}{9}=55.556[/tex]For k = 6.1
[tex]V_{new}=(12^{\frac{1}{3}}\times6.1)^3_{}=2723.772[/tex]Note that all answers are in cubic units.
