What is the volume of the largest box that can be made from a square piece of cardboard with side lengths of 24 inches by cuttin g equal squares from each corner and turning up the sides?
2 answers:
Answer:
1024 in³
Step-by-step explanation:
Volume = (24 - 2x)(24 - 2x)(x)
= 576x - 96x² + 4x³
dV/dx = 576 - 192x + 12x² = 0
x² - 16x + 48 = 0
x² - 12x - 4x + 48 = 0
x(x - 12) - 4(x - 12) = 0
(x - 12)(x - 4) = 0
x = 4, x = 12
x can not be 12,
Because 24 - 2(12) = 0
So x = 4
Volume = (24 - 8)(24 - 8)(4)
= 1024
You need to subtract the two cutouts on each side, so the side of the box would be 24- 2x. X would then become the height of the box once folded.
The volume is found by multiplying the 4 dimensions:
The volume becomes x(24-2x)^2
Now take the derivative of the formula to solve for x
Dv/dx= (24-2x)^2 + x^2(24-2x)*-2
= 24-2x(24-2x-4x)=0
X =12 or 4
24-2x = 24-2(4) = 24-8=16
Volume = 16 x 16 x 4 = 1024 cubic inches.
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