Two similar rectangular prisms have a scale factor of 4:1. The smaller prism has a volume of 6 cubic centimeters. What is the volume of the larger prism in cubic centimeters?
the volume ratio is the cube (3rd power) of the scale factor
To find the volume of the larger prism, we can use the concept of scale factor. The scale factor describes the relationship between the two prisms and can be used to find the corresponding measurements.
In this case, the scale factor is given as 4:1. This means that the measurements of the larger prism are four times greater than the measurements of the smaller prism. Since we are looking for the volume, which is a three-dimensional measurement, we need to consider the scale factor in all three dimensions (length, width, and height).
Let's assume the volume of the larger prism is V cubic centimeters.
Since the scale factor is 4:1, the height of the larger prism is four times the height of the smaller prism (h₁).
So, the height of the larger prism is 4h₁.
The volume of a rectangular prism is calculated by multiplying the length, width, and height. We know the volume of the smaller prism is 6 cubic centimeters, so we can set up the equation:
length₁ × width₁ × height₁ = 6
Since the scale factor is 4:1, the length and width of the larger prism are also four times the length and width of the smaller prism. So, we have:
(4×length₁) × (4×width₁) × (4×height₁) = V
We can simplify this equation as:
64×length₁×width₁×height₁ = V
But we know that length₁ × width₁ × height₁ = 6. So, we can substitute this value in the equation:
64×6 = V
V = 384 cubic centimeters
Therefore, the volume of the larger prism is 384 cubic centimeters.