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What is the ratio for the surface areas of the rectangular prisms shown below, given that they are similar and that the ratio of their edge lengths is 8:5?

User Anayansi
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1 Answer

1 vote

Answer:

64:25

Explanation:

It is given that the ratio of their edge lengths is 8:5.

Therefore,
(l_(1) )/(l_(2)) =(b_(1) )/(b_(2)) =(h_(1) )/(h_(2)) =(8)/(5)

where
l_(1) ,b_(1) and
h_(1) are the length, breadth and height of the first prism and
l_(2) ,b_(2) and
h_(2) are the length, breadth and height of the second prism.

So,


l_(1) =(8)/(5) l_(2)


b_(1) =(8)/(5) b_(2) and


h_(1) =(8)/(5) h_(2)

Now, surface area of a rectangular prism is :

A = 2(lb + bh + hl)

Therefore, ratio of the surface areas is:


(A_(1) )/(A_(2)) =(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) ))


=(l_(1)b_(1)+b_(1)h_(1)+h_(1)l_(1) )/(l_(2)b_(2)+b_(2)h_(2)+h_(2)l_(2) )


=(((8)/(5) l_(2) )((8)/(5) b_(2) )+((8)/(5) b_(2) )((8)/(5) h_(2) )+((8)/(5) h_(2) )((8)/(5) l_(2) ))/(l_(2)b_(2)+b_(2)h_(2)+h_(2)l_(2) )


=((64)/(25) (l_(2)b_(2) )+(64)/(25) (b_(2)h_(2) )+(64)/(25) (h_(2)l_(2) ))/(l_(2)b_(2)+b_(2)h_(2)+h_(2)l_(2) )


=((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) )


=(64)/(25)

Hence, the ratio of the surface areas of the rectangular prisms is 64:25.

User Leo Selig
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7.0k points