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Radda [10]
2 years ago
5

A box is to be constructed from a sheet of cardboard that is 10 cm by 60 cm by cutting out squares of length x by x from each co

rner and bending up the sides. What is the maximum volume this box could have?
Mathematics
1 answer:
Stells [14]2 years ago
6 0

Volume of a rectangular box = length x width x height<span>

From the problem statement,
length = 60 - 2x
width = 10 - 2x
height = x</span>

<span>
where x is the height of the box or the side of the equal squares from each corner and turning up the sides

V = (60-2x) (10-2x) (x)
V = (60 - 2x) (10x - 2x^2)
V = 600x - 120x^2 -20x^2 + 4x^3
V = 4x^3 - 100x^2 + 600x

To maximize the volume, we differentiate the expression of the volume and equate it to zero.

V = </span>4x^3 - 100x^2 + 600x<span>
dV/dx = 12x^2 - 200x + 600
12x^2 - 200x + 600 = 0</span>

<span>x^2 - 50/3x + 50 = 0

Solving for x,
x1 = 12.74 ;      Volume = -315.56  (cannot be negative)
x2 = 3.92 ;      Volume = 1056.31

So, the answer would be that the maximum volume would be 1056.31 cm^3.
</span><span>
</span>
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