An open-top box is to be made by cutting small congruent squares from the corners of a 9 cm × 9 cm sheet of metal and bending up
the sides. what is largest possible volume of such a box
1 answer:
Let the length of the side of the 4 small squares be = x.
The formula for the volume of the box will be
height * width * length
V = x ( 9 - 2x)^2
V = 81x - 36x^2 + 4x^3
finding the derivative:-
dV / d x = 12x^2 - 72x + 81
THis equals 0 for a maximum / minimum value
12x^2 - 72x + 81 = 0
3(4x^2 - 24x + 27) = 0
x = 4.5 , 1.5
Use second derivative to find maxm and minm:-
d^2V / dx^2 = 24x - 72
when x = 1.5 this is negative and when x = 4.5 this is positive
so x = 1.5 gives a maximum value for V
V = 1.5(9- 2(1-5))^2 = 54
Largest possible volume of the box is 54 cm^3 Answer
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