Asked by Paige
What is the maximum volume of a closed box with a square base which can be made by bending the material? The box is to have a surface area of 100in^2.
Answers
Answered by
Steve
If the final box has bases x^2 and height h, then
v = x^2 h
and we also have
x^2 + 4xh = 100, so
h = (100-x^2)/4x
That means
v = x^2 (100-x^2)/4x
= x/4 (100-x^2)
= 25x - 1/4 x^3
dv/dx = 25 - 3/4 x^2
maximum volume is achieved when dv/dx=0.
v = x^2 h
and we also have
x^2 + 4xh = 100, so
h = (100-x^2)/4x
That means
v = x^2 (100-x^2)/4x
= x/4 (100-x^2)
= 25x - 1/4 x^3
dv/dx = 25 - 3/4 x^2
maximum volume is achieved when dv/dx=0.
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