Question

A solid with a volume of 12 cubic units is dilated by a scale factor of K. What is the volume of the solid when K equals: 1/40.411.25/36.1

Answer 1

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 :

`V_{\text{new}}=\lbrack(V^{}_{\text{orig}})^{(1)/(3)}* k\rbrack^3`

The original volume must be raised to 1/3 to get one unit dimension..

`(units^3)^{(1)/(3)}=unit`

Substitute the original volume and the values of k to the formula :

For k = 1/4

`V_(new)=(12^{(1)/(3)}*(1)/(4))^3_{}=(3)/(16)=0.188`

For k = 0.4

`V_(new)=(12^{(1)/(3)}*0.4)^3_{}=(9)/(125)=0.072`

For k = 1

Volume will still be the same since the scale factor is 1.

For k = 1.2

`V_(new)=(12^{(1)/(3)}*1.2)^3_{}=(2592)/(125)=20.736`

For k = 5/3

`V_(new)=(12^{(1)/(3)}*(5)/(3))^3_{}=(500)/(9)=55.556`

For k = 6.1

`V_(new)=(12^{(1)/(3)}*6.1)^3_{}=2723.772`

Note that all answers are in cubic units.