We are given the dimensions of the box:
l = 50 cm
w = 20 cm
We know that the volume of a box is:
V = l w h
where h = x
However since x cuts is made on both all sides of the box,
therefore the new dimensions would be:
V = (l – 2x) (w – 2x) x
V = (50 – 2x) (20 – 2x) x
V = 1000x – 100x^2 – 40x^2 + 4x^3
V = 4x^3 – 140x^2 + 1000x
<span>To get the maxima value, we get the 1st
derivative of the function then set dV/dx = 0 to solve for x:</span>
dV / dx = 12x^2 – 280x + 1000
12x^2 – 280x + 1000 = 0
Transpose 1000 to the right side and divide everything by
12:
x^2 – (280/12)x = -(1000/12)
Completing the square:
x^2 – (280/12)x + (78400/576) = -(1000/12) + (78400/576)
[x – (280/24)]^2 = 52.78
x – (280/24) = ±7.26
<span>x = (280/24) ± 7.26</span>
x = 4.40, 18.93
x cannot be 18.93 since this would result in a negative
value of 20 – 2x, therefore:
x = 4.40 cm
Calculating for the volume:
V = (50 – 2*4.4) (20 – 2*4.4) (4.4)
<span>V = 2030.34 cm^3</span>
