An open box is to be made out of a 12-inch by 18-inch piece by cutting out squares of equal size from the four corners and bending up the sides.

Question

Find the dimensions of the resulting box that has the largest volume

dimensions???

Height??

I think you have to use optimization

Answer 1

Let x is side of squares. Then volume v = x(12 - 2x)(18 - 2x); v = 4x(6 - x)(9 - x) = 4(x³ - 15x² + 54x);

v' = 4(3x² - 30x + 54) = 0; x² - 10x + 18 = 0; x = 5 + √7 > 6 (rejecting) or x = 5 - √7

v'' = 4(6x - 30); v''(5 - √7) = - 24√7 < 0 (max)

So, (5 - √7)x(2 + 2√7)x(8 + 2√7)