Suppose you have two similar rectangular prisms. The volume of the smaller rectangular prism is 125 in3 and the volume of the largest rectangular prism is 729 in3. What is the scale factor of the smaller figure to the larger figure?
To find the scale factor, we need to compare the dimensions (length, width, and height) of the two prisms.
Let's call the dimensions of the smaller prism l, w, and h. Then we know that:
Volume of the smaller prism = l × w × h = 125
Next, let's call the dimensions of the larger prism kl, kw, and kh (where k is the scale factor we're trying to find). Then we know that:
Volume of the larger prism = kl × kw × kh = 729
To get from the smaller prism to the larger prism, each dimension is multiplied by the same scale factor k. So we can set up the following equation:
k × l × k × w × k × h = 729
Simplifying this equation, we get:
k^3 × lwh = 729
We can substitute the volume of the smaller prism into this equation:
k^3 × 125 = 729
Solving for k, we get:
k = (729/125)^(1/3)
k ≈ 1.5
So the scale factor from the smaller prism to the larger prism is approximately 1.5. This means that each dimension of the larger prism is 1.5 times larger than the corresponding dimension of the smaller prism.
To find the scale factor of the smaller rectangular prism to the larger rectangular prism, we can compare their volumes.
The volume of the smaller rectangular prism is 125 in³, and the volume of the larger rectangular prism is 729 in³.
To find the scale factor, we divide the volume of the larger prism by the volume of the smaller prism:
729 in³ / 125 in³ = 5.832
Therefore, the scale factor of the smaller rectangular prism to the larger rectangular prism is approximately 5.832.