Mensa Box Problem Solution

This problem especially tests one's Algebra 2 facility in radicals, algebraic fractions & exponents. Three recent posts for which appeared on the blog.

The  area of the box floor is:

(x  -   2y) ²   = x ²  - 4 xy  +   4y ²   

 The volume of the box is: 

y(x  -   2y) ²   = x ² y  - 4 xy+   4y³   

The maximum of a function is found by calculating the derivative, setting it to zero and finding the critical points. The derivative of the volume function with respect to y is:

x ² y  - 4 xy+   4y³    = x ²  - 8 xy  +   12y ²  

=  12y ²  - 8 xy  + x ²

Apply the quadratic formula:

x = -b + Ö {b² - 4ac}/ 2a

Where: b = 8x, c = 1, a = 12

So: 

y =  8 x + Ö {64 x² - 48x² }/ 24

^{y =  8 x + Ö {16x2 }/ 24}

y =  8 x + 4 x / 24

y = x/ 2,  x/ 6

But y cannot equal x/2 or the volume would be zero.

Therefore, the volume of the box is a maximum when y = x/ 6:

Using the volume eqn.: y(x  -   2y) ²

Þ 

(x/6) {x - 2 (x/6)²} =

In 2nd term in parentheses change: 2(x/6) = 2x/ 6 = x/3

Subst. back to get:   x -  x/3 = 3x/ 3 - x/ 3 =  2x/ 3

Square this term inside parentheses: (2x/3)² =  4x²/9

Multiply now by external factor (x/6):

(x/6)  {4x²/9 } =   4x³ /54