Let A and B be two rectangular prisms which are proportional.
The dimensions of rectangular prism A:
Length, l=10 cm.
Width, w=5 cm.
Height, h=3 cm.
The dimensions of rectangular prism B:
Length, L=20 cm.
Width, w=10 cm.
Height, h=6 cm.
Two prisms are proportional if the ratio of the corresponding dimensions of the prisms are equal.
The ratio of corresponding dimensions of prism A to B is,
[tex]\begin{gathered} \frac{l}{L}=\frac{10\text{ cm}}{20\text{ cm}}=\frac{1}{2} \\ \frac{w}{W}=\frac{5\text{ cm}}{10\text{ cm}}=\frac{1}{2} \\ \frac{h}{H}=\frac{3\text{ cm}}{6\text{ cm}}=\frac{1}{2} \end{gathered}[/tex]Since the ratios of lengths, widths and heights are equal, the prisms A and B are proportional.
Now, the volume of rectangular prism A is,
[tex]\begin{gathered} V_A=\text{lwh} \\ =10\times5\times3 \\ =150cm^3_{\rbrack} \end{gathered}[/tex]The surface area of rectangular prism A is,
[tex]\begin{gathered} S_A=2(lw+lh+wh) \\ =2(10\times5+10\times3+5\times3) \\ =2(50+30+15) \\ =2\times95 \\ =190cm^2 \end{gathered}[/tex]Therefore, the volume of rectangular prism A is 150 cu.cm and the surface area rectangular prism A is 190 sq.cm.
The volume of prism B is,
[tex]\begin{gathered} V_B=LWH \\ =20\times10\times6 \\ =1200cm^3 \end{gathered}[/tex]The surface area of prism B is,
[tex]\begin{gathered} S_B=2(LW+LH+WH) \\ =2(20\times10+20\times6+10\times6) \\ =2(200+120+60) \\ =2\times380 \\ =760cm^2 \end{gathered}[/tex]Therefore, the volume of rectangular prism A is 1200 cu.cm and the surface area rectangular prism A is 760 sq.cm.
A rought sketch of the prism is shown below:
