Questions: a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 39 ft by 21 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Suppose that in part (a) the original piece of cardboard is a square with sides of length s. Find the volume of the largest box that can be formed in this way. a. To find the objective function, express the volume V of the box in terms of x. V = (Type an expression.)

a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 39 ft by 21 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.
b. Suppose that in part (a) the original piece of cardboard is a square with sides of length s. Find the volume of the largest box that can be formed in this way.
a. To find the objective function, express the volume V of the box in terms of x.

V =
(Type an expression.)

Transcript text: a. Squares with sides of length $x$ are cut out of each corner of a rectangular piece of cardboard measuring 39 ft by 21 ft . The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Suppose that in part (a) the original piece of cardboard is a square with sides of length s . Find the volume of the largest box that can be formed in this way. a. To find the objective function, express the volume $V$ of the box in terms of $x$. \[ \mathrm{V}=\square \] (Type an expression.)

Solution

Solution Steps

Step 1: Derive the volume formula

The volume of the box, $V = (l - 2x)(w - 2x)x$, where $l = 39$ and $w = 21$.

Step 2: Differentiate the volume formula with respect to $x$ and find critical points

The derivative of $V$ with respect to $x$ is $-2_x_(21 - 2_x) - 2_x_(39 - 2_x) + (21 - 2_x)_(39 - 2*x)$, and solving $dV/dx = 0$ gives the critical points.

Step 3: Determine the maximum volume

Substitute the critical points back into the volume formula and verify the second derivative test. The maximum volume is found to be approximately 1621.22 cubic units.

Final Answer:

The maximum volume of the box that can be formed is approximately 1621.22 cubic units.

Step 1: Derive the volume formula

The volume of the box, $V = (s - 2x)^2x$, where $s = 39$.

Step 2: Differentiate the volume formula with respect to $x$ and find critical points

The derivative of $V$ with respect to $x$ is $x_(8_x - 156) + (39 - 2*x)^2$, and solving $dV/dx = 0$ gives the critical points.

Step 3: Determine the maximum volume

Substitute the critical points back into the volume formula and verify the second derivative test. The maximum volume is found to be approximately 4394 cubic units.

Final Answer:

The maximum volume of the box that can be formed is approximately 4394 cubic units.

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