Question

The perimeter of base dhgc is 30 inches, and the perimeter of base swvr is 20 inches. Complete the table with the ratios of the heights, surface areas, and volumes of these two similar rectangular prisms. Assume the ratios are written in the form figure 1 : figure 2.

Answer 1

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

$p' = k_{p}\cdot p$ (1)

Surface area

$A'_{s} = k_{A_{s}}\cdot A_{s}$ (2)

Volume

$V' = k_{V}\cdot V$ (3)

Where:

$p$, $p'$ - Perimeters of the small cuboid and the big cuboid, in inches.

$A_{s}$, $A'_{s}$ - Surface areas of the small cuboid and the big cuboid, in square inches.

$V$, $V'$ - 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

$4\cdot (w' + h' + l') = k_{p}\cdot [4\cdot (w + h + l)]$

$4\cdot (k\cdot w'+ k\cdot h' + k\cdot l) = k_{p}\cdot [4\cdot (w+h+l)]$

$k_{p} = k$

Surface area

$2\cdot (w'\cdot h' + l'\cdot w' + l' \cdot h') = k_{A_{s}}\cdot [2 \cdot(w\cdot h + l \cdot w + l\cdot h)]$

$2\cdot [(k\cdot w')\cdot (k\cdot h') + (k\cdot l)\cdot (k\cdot w) + (k\cdot l)\cdot (k\cdot h)] = k_{A_{s}}\cdot [2\cdot (w\cdot h + l\cdot w + l\cdot h)]$

$k_{A_{s}} = k^{2}$

Volume

$w'\cdot h' \cdot l' = k_{V}\cdot (w\cdot h \cdot l)$

$(k\cdot w)\cdot (k \cdot h)\cdot (k \cdot l) = k_{V}\cdot (w\cdot h \cdot l)$

$k_{V} = k^{3}$

Where $k = \frac{p'}{p}$.

If we know that $p' = 30\,in$ and $p = 20\,in$, then we proceed to calculate all the ratios:

$k_{p} = \frac{30\,in}{20\,in}$

$k_{p} = \frac{3}{2}$

$k_{A_{s}} = \frac{9}{4}$

$k_{V} = \frac{27}{8}$