A and B are two solid cuboids that are mathematically similar. The surface area scale factor from A to B is 3.24. Find the ratio of the volume of cuboid A to the volume of cuboid B.
Answer (1)
When dealing with similar three-dimensional shapes like cuboids, understanding the relationships between surface area and volume is important.
Surface Area Scale Factor : The problem states that the surface area scale factor from cuboid A to cuboid B is 3.24.
Finding the Side Length Scale Factor : If two shapes are similar, the scale factor (ratio) for their sides, surface areas, and volumes have specific relationships. For two similar solids:
The scale factor of corresponding lengths can be found by taking the square root of the surface area scale factor. Therefore, the scale factor for the lengths is:
k = 3.24 = 1.8
Volume Scale Factor : The volume of similar shapes is proportional to the cube of the side length scale factor. So, the volume scale factor is:
k 3 = ( 1.8 ) 3 = 5.832
Ratio of Volumes : From cuboid A to cuboid B, the volume ratio will be:
Volume Ratio from A to B = 5.832 1
Since we were tasked to find the ratio of the volume of cuboid A to cuboid B, the resulting ratio is:
1 : 5.832
Therefore, the ratio of the volume of cuboid A to the volume of cuboid B is 1:5.832, meaning that for every unit volume of A, B has 5.832 units of volume.
