Question

a manufacturer wants to design an open box that has a square base and a surface area of 48sq units what dimension will produce a box with a maximum volume?

Answer 1

Let the side be x and the height h. Then x^2+4xh=48 and h=(48-x^2)/(4x). Now we have an expression for the volume in x: V(x)=x^2h=x^2*(48-x^2)/(4x)=12x-x^3/4. Taking the derivative we get V'(x)=12-3/4x^2=0. The volume is maximal when V'(x)=0 so 12-3/4x^2 and the positive solution is x=4. So h=(48-16)/16=2. We verify that is a maximum point by considering V''(x)=-3/2x so V''(2)=-3<0 and hence a mximum point. The dimension of the box is 4x4x2 units.

Answer 2

maths

Answer 3

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Answer 4

32