Posted by sadia khan on Saturday, January 19, 2013 at 2:42am.
let each side of the cut-out square be x cm
where 0 < x < 12
base of box is (24-2x) by (45-2x)
height will be x
volume = x(24-2x)(45-2x)
= x(1080 - 138x + 4x^2) = 4x^3 - 138x^2 + 1080x
d(volume)/dx = 12x^2 - 276x + 1080
= 0 for a max/min of volume
x^2 - 23x + 90 = 0
(x-18)(x-5) = 0
x = 18 or x = 5
but x < 12 , so x = 5
a square of 5 by 5 cm should be cut out.
test:
if x = 4.8
volume = 4.8(24-9.6)(45-9.6) = 2446.848
if x = 5
volume = 5(24-10)(45-10) = 24050
if x = 5.1
volume = 5.1(24-10.2)(45-10.2) = 2449.224
max when x=5
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