Answer:
Ratio of heights: 3 : 2
Ratio of surface areas: 9 : 4
Ratio of volumes: 27 : 8
Step-by-step explanation:
Since both cuboids are proportional, then we must derive expressions for the ratios of heights, surface areas and volumes from the following identities:
Perimeter
[tex]p' = k_{p}\cdot p[/tex] (1)
Surface area
[tex]A'_{s} = k_{A_{s}}\cdot A_{s}[/tex] (2)
Volume
[tex]V' = k_{V}\cdot V[/tex] (3)
Where:
[tex]p[/tex], [tex]p'[/tex] - Perimeters of the small cuboid and the big cuboid, in inches.
[tex]A_{s}[/tex], [tex]A'_{s}[/tex] - Surface areas of the small cuboid and the big cuboid, in square inches.
[tex]V[/tex], [tex]V'[/tex] - Volumes of the small cuboid and the big cuboid, in cubic inches.
By means of geometry formulas we expand the system of equations below:
Perimeter
[tex]4\cdot (w' + h' + l') = k_{p}\cdot [4\cdot (w + h + l)][/tex]
[tex]4\cdot (k\cdot w'+ k\cdot h' + k\cdot l) = k_{p}\cdot [4\cdot (w+h+l)][/tex]
[tex]k_{p} = k[/tex]
Surface area
[tex]2\cdot (w'\cdot h' + l'\cdot w' + l' \cdot h') = k_{A_{s}}\cdot [2 \cdot(w\cdot h + l \cdot w + l\cdot h)][/tex]
[tex]2\cdot [(k\cdot w')\cdot (k\cdot h') + (k\cdot l)\cdot (k\cdot w) + (k\cdot l)\cdot (k\cdot h)] = k_{A_{s}}\cdot [2\cdot (w\cdot h + l\cdot w + l\cdot h)][/tex]
[tex]k_{A_{s}} = k^{2}[/tex]
Volume
[tex]w'\cdot h' \cdot l' = k_{V}\cdot (w\cdot h \cdot l)[/tex]
[tex](k\cdot w)\cdot (k \cdot h)\cdot (k \cdot l) = k_{V}\cdot (w\cdot h \cdot l)[/tex]
[tex]k_{V} = k^{3}[/tex]
Where [tex]k = \frac{p'}{p}[/tex].
If we know that [tex]p' = 30\,in[/tex] and [tex]p = 20\,in[/tex], then we proceed to calculate all the ratios:
[tex]k_{p} = \frac{30\,in}{20\,in}[/tex]
[tex]k_{p} = \frac{3}{2}[/tex]
[tex]k_{A_{s}} = \frac{9}{4}[/tex]
[tex]k_{V} = \frac{27}{8}[/tex]
