Answer:
64:25
Step-by-step explanation:
It is given that the ratio of their edge lengths is 8:5.
Therefore, [tex]\frac{l_{1} }{l_{2}} =\frac{b_{1} }{b_{2}} =\frac{h_{1} }{h_{2}} =\frac{8}{5}[/tex]
where [tex]l_{1} ,b_{1}[/tex] and [tex]h_{1}[/tex] are the length, breadth and height of the first prism and [tex]l_{2} ,b_{2}[/tex] and [tex]h_{2}[/tex] are the length, breadth and height of the second prism.
So,
[tex]l_{1} =\frac{8}{5} l_{2}[/tex]
[tex]b_{1} =\frac{8}{5} b_{2}[/tex] and
[tex]h_{1} =\frac{8}{5} h_{2}[/tex]
Now, surface area of a rectangular prism is :
A = 2(lb + bh + hl)
Therefore, ratio of the surface areas is:
[tex]\frac{A_{1} }{A_{2}} =\frac{2(l_{1}b_{1}+b_{1}h_{1}+h_{1}l_{1} )}{2(l_{2}b_{2}+b_{2}h_{2}+h_{2}l_{2} )}[/tex]
[tex]=\frac{l_{1}b_{1}+b_{1}h_{1}+h_{1}l_{1} }{l_{2}b_{2}+b_{2}h_{2}+h_{2}l_{2} }[/tex]
[tex]=\frac{(\frac{8}{5} l_{2} )(\frac{8}{5} b_{2} )+(\frac{8}{5} b_{2} )(\frac{8}{5} h_{2} )+(\frac{8}{5} h_{2} )(\frac{8}{5} l_{2} )}{l_{2}b_{2}+b_{2}h_{2}+h_{2}l_{2} }[/tex]
[tex]=\frac{\frac{64}{25} (l_{2}b_{2} )+\frac{64}{25} (b_{2}h_{2} )+\frac{64}{25} (h_{2}l_{2} )}{l_{2}b_{2}+b_{2}h_{2}+h_{2}l_{2} }[/tex]
[tex]=\frac{\frac{64}{25} (l_{2}b_{2}+b_{2}h_{2}+h_{2}l_{2} )}{l_{2}b_{2}+b_{2}h_{2}+h_{2}l_{2} }[/tex]
[tex]=\frac{64}{25}[/tex]
Hence, the ratio of the surface areas of the rectangular prisms is 64:25.
