Please solve A box with a square base and open top must have a volume of 256000 cm³. We wish to find the dimensionsof the box that minimize the amount of material used.First, find a formula for the surface area of the box in terms of only x, the length of one side of the squarebase.[Hint: use the volume formula to express the height of the box in terms of x.]Simplify your formula as much as possible.A(x) =Next, find the derivative, A'(x).= (2),VNow, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by a?A'(x) = 0 when x =We next have to make sure that this valuesecond derivative test. Find A"(x).x gives a minimum value for the surface area. Let's use theA"(x)Evaluate A"(x) at the x-value you gave above.NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around thatvalue, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.)

We use second derivative test to find dimensions where surface Area minimum

A box with a square base and open top must have a volume of 256000 cm. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible. A(x): Next, find the derivative, A'(x). A' (x) = %3D Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by a2 A'(x) = 0 when x = We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) = Evaluate A"(x) at the x-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that minimum for 4(m) (Your hoss is hanny now.)