There are two cubes. Their lengths are in the ratio 3:4, what is the simplified ratio between their volumes?
a1 = side length of first cube
a2 = side length of second cube
V1 = volume of first cube
V2 = volume of second cube
a1 / a2 = 3 / 4 Multiply both sides by a2
a1 = 3 a2 / 4 = ( 3 / 4 ) a2
V1 / V2 = a1³ / a2³ =
[ ( 3 / 4 ) a2 ]³ / a2³ =
( 27 / 64 ) a2³ / a2³ = 27 / 64
To find the simplified ratio between the volumes of two cubes, we can use the relationship between the lengths and the volumes of cubes.
The length of a side of a cube represents one dimension, so the ratio of the lengths of the sides of the two cubes will give us the ratio of their volumes. In this case, the lengths of the sides are in the ratio of 3:4.
Let's assume the length of the side of the first cube is 3x, where x is a common factor. Therefore, the volume of the first cube would be (3x)^3 = 27x^3.
Similarly, the length of the side of the second cube is 4x, and its volume would be (4x)^3 = 64x^3.
Now, to find the simplified ratio, we divide the volume of the second cube by the volume of the first cube: 64x^3 / 27x^3.
Dividing both terms by the common factor, x^3, we get: 64/27.
Hence, the simplified ratio between the volumes of the two cubes is 64:27.