By what factor should each edge of a cube be increased in order to increase its volume by a factor of 27?
a.) 27 b.) 9 c.) 3 d.) 2
if the edge changes from x to nx, then
the volume changes from x^3 to (nx)^3, or n^3 * x^3
so, if n^3 * x^3 = 27*x^3, n^3 = 27, so n=3
Just remember:
If the linear dimension changes by a factor of n,
area grows by n^2
volume grows by n^3
To find the answer to this question, we need to understand the relationship between the volume of a cube and the length of its edges.
The volume of a cube is given by the formula V = a^3, where "a" represents the length of each edge.
We are given that the volume should be increased by a factor of 27.
By substituting the formula for volume into the equation, we can set up the following equation:
27V = (27a)^3
Next, we want to isolate "a" in order to determine by what factor each edge should be increased.
Dividing both sides of the equation by 27, we get:
V = a^3
(27V)/27 = (27a)^3/27
V = (a/3)^3
Now we can see that in order to increase the volume by a factor of 27, we need to increase each edge of the cube by a factor of 3.
Therefore, the correct answer is c.) 3.