SOLUTION: can you help me with this problem. What is the maximum possible volume of rectangular box whose longest diagonal has fixed length L. thank you for helping me.

Question 361173: can you help me with this problem. What is the maximum possible volume of rectangular box whose longest diagonal has fixed length L. thank you for helping me.

Answer by edjones(8007) (Show Source): You can put this solution on YOUR website!
A square box has the largest volume.
x=length of each side of the square base of the box and the height of the box also, d=diagonal of base of box, L=diagonal of the box from bottom corner to opposite top corner.
d=xsqrt(2) hypotenuse of isosceles right triangle.
L^2=h^2+d^2
=x^2*(xsqrt(2))^2
=x^2*2x^2
=2x^4
L=x^2sqrt(2)
.
V=x^3 given
Ly=x^3 To find volume in terms of L we introduce a new variable, y, which, when multiplied by L equals the volume.
x^2sqrt(2)y=x^3 Substitute x^2sqrt(2) for L
y=x^3/x^2sqrt(2)
=x/sqrt(2)
=x/sqrt(2) * sqrt(2)/sqrt(2) Rationalizing the denominator.
=x*sqrt(2)/2
V=(L*x*sqrt(2))/2
.
Ed

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