Question

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?

Answer 1

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.