Explanation
1. Define the Variables
Let the side of the square cut from each corner be x. This x will also be the height of the open-top box.
2. Dimensions of the Box
After cutting squares of side x from the corners of a 6m×6m sheet:
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Length (l): 6−2x
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Width (w): 6−2x
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Height (h): x
3. Formulate the Volume Equation
The volume V of the box is given by:
4. Find the Derivative for Maxima
To maximize volume, find dxdV and set it to zero:
Set 12x2−48x+36=0:
So, x=1 or x=3.
5. Determine the Valid Height
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If x=3, the length 6−2(3)=0, which is impossible.
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If x=1, the dimensions are 4m×4m×1m, which is valid.
To confirm maximum volume, check the second derivative dx2d2V=24x−48.
For x=1, 24(1)−48=−24 (Negative, so it is a Maxima).
The required height is 1 m. Since 1 m is not among the options (a), (b), or (c):
Correct Option: (d) None
