Calculating Weight Savings with Rectangular Section Beam

[MMCS]

Here is the original question, i know the first answer as 182mm

A square section cantilever beam, 3 m in length, carries a concentrated load of 10 kN at its free end. If the maximum bending stress is not to exceed 30 MPa determine the minimum dimensions of the section.

Ans 182mm,

What would be the percentage saving in weight if a rectangular section beam, having a depth equal to twice its width, were to be used instead?

Intuitively it seems there would be a weight reduction of 50% if the width dimension was halved however I have the answer to be 29.6%, does somebody know how i would get to this?

Thanks

 

[SteamKing]

Why don't you show some calculations and see if the answer pops out.

 

[haruspex]

Intuitively it seems there would be a weight reduction of 50% if the width dimension was halved

Equally intuitively, it would have half the load bearing capability. There would have to be a corresponding increase in depth.

 

[MMCS]

How would i use this in the fomula? To solve the first question i used I to be x*x^3/12, but now because it is of rectangular cross section i would use b and d, however this gives me two uknowns

 

[SteamKing]

You know a relationship between breadth and depth. From the OP, assume D = 2*B

 

[MMCS]

Here is my working, i am a few percent off, can you see any mistakes?