# (Solved): Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a=8 inch ...

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

: (6) The value of

V

at the left endpoint is (7) The value of

V

at the right endpoint is (8) The maximum volume is

V=

(9) Answer the original question. The value of

x

that maximizes the volume is:

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