Questions: a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 33 ft by 18 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. The maximum volume of the box is approximately ft^3. (Round to the nearest hundredth as needed.)

a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 33 ft by 18 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. The maximum volume of the box is approximately ft^3. (Round to the nearest hundredth as needed.)
Transcript text: a. Squares with sides of length $x$ are cut out of each corner of a rectangular piece of cardboard measuring 33 ft by 18 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. The maximum volume of the box is approximately $\square$ $\mathrm{ft}^{3}$. (Round to the nearest hundredth as needed.)
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Solution

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Solution Steps

Step 1: Derive the volume formula

The volume of the box, $V = (l - 2x)(w - 2x)x$, where $l = 33$ and $w = 18$.

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

The derivative of $V$ with respect to $x$ is $-2_x_(18 - 2_x) - 2_x_(33 - 2_x) + (18 - 2_x)_(33 - 2*x)$, 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 1004.08 cubic units.

Final Answer:

The maximum volume of the box that can be formed is approximately 1004.08 cubic units.

Step 1: Derive the volume formula

The volume of the box, $V = (s - 2x)^2x$, where $s = 33$.

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

The derivative of $V$ with respect to $x$ is $x_(8_x - 132) + (33 - 2*x)^2$, 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 2662 cubic units.

Final Answer:

The maximum volume of the box that can be formed is approximately 2662 cubic units.

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