Open-box Problem. An open-box (top open) is made from a...

Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a = 8 inches by b = 7 inches by cutting a square of side $x$ at each corner and turning up the sides (see the figure). Determine the value of $x$ that results in a box the maximum volume. Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume $V$ as a function of $x$: $V =$ (2) Determine the domain of the function $V$ of $x$ (in interval form): (3) Expand the function $V$ for easier differentiation: $V =$ (4) Find the derivative of the function $V$: $V' =$ (5) Find the critical point(s) in the domain of $V$:

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A box with a square base and open top must have a volume of 500000cm^(3). 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)= Now, calculate when the derivative equals zero, that is, when A^(')(x)=0. [Hint: multiply both sides by x^(2) .] A^(')(x)=0 when x= We next have to make sure that this value of x 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 value, so the zero of A^(')(x) must indicate a local minimum for A(x). (Your boss is happy now.) Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a=8 inches by b=7 inches by cutting a square of side x at each corner and turning up the sides (see the figure). Determine the value of x that results in a box the maximum volume. Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume V as a function of x:V= (2) Determine the domain of the function V of x (in interval form): (3) Expand the function V for easier differentiation: V= (4) Find the derivative of the function V:V^(')= (5) Find the critical point(s) in the domain of V :

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