Question

Consider a rectangular prism with a total surface area of 94 in^2 . If the sum of all of its edges is 48 in, what is the sum of the lengths of all of its interior diagonals, in inches? Based on the given information, can we determine the exact dimensions of the prism? Why or why not?

Answer 1

`20\sqrt5`

Step-by-step explanation:

According to the question

`2lb+2bh+2hl=94`

`4l+4b+4h=48\\\Rightarrow 4(l+b+h)=48\\\Rightarrow l+b+h=12`

Length of the diagonal is given by

`d=√(l^2+b^2+h^2)`

This can be also written as

`d=√((l+b+h)^2-(2lb+2bh+2hl))\\\Rightarrow d=√(12^2-(94))\\\Rightarrow d=√(144-94)\\\Rightarrow d=√(50)=5\sqrt2\ in`

The length of one diagonal is
`5\sqrt2\ in`

As there are 4 diagonals the sum of the lengths of the prism is
`4* 5\sqrt2=20\sqrt5`

With the given information the exact dimensions of the prism cannot be determined as the two equations cannot be solved and trigonometry can also be not used.