A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm2) is equal to :
Explanation
Step 1: Volume Maximization
The volume V of the box is given by:
V=(900−120x+4x2)x=4x3−120x2+900x
To find the maximum volume, we take the derivative with respect to x and set it to zero:
Dividing the equation by 12:
The possible values for x are 5 or 15.
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If x=15, the length 30−2(15)=0, which is impossible.
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Therefore, x=5 cm for maximum volume.
(Checking the second derivative: dx2d2V=24x−240. At x=5, 120 - 240 = -120 < 0, confirming a maximum).
Step 2: Calculating Surface Area
The box is an open box (no top). The surface area S is the area of the base plus the area of the four side walls.
The dimensions at x=5 are:
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Length l=30−2(5)=20 cm
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Width w=30−2(5)=20 cm
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Height h=5 cm
The surface area S is:
S=Area of base+4×(Area of one side)
Final Answer:
The surface area of the box is 800 cm².
