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A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sides
What is the value of x for width the vulume is maximum ?
  • a)
    1 inch
  • b)
    1.5 inch
  • c)
    2 inch
  • d)
    2.5 inch
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
A rectangular box is to be made from a sheet of 24 inch length and 9 i...

length o f the box = 24 - 2x
width of the box = 9 - 2x
Volume of the box=V
V = (24 - 2x ) (9 -2x ) .x (∵height o f box = x inch)
= ( 216 - 48 x - 18x +4x2).x
V (x)= 4x3 - 66x2 + 2 16x
⇒ V'(x)= 12x2-132x+ 216 For maximum, put V'(x)=0
⇒ 12x2- 132x+216=0
⇒x2 - l l x + 18 = 0
⇒x2 - 9x - 2x + 18 = 0
⇒ x (x -9)-2(x-9)= 0
⇒ (x -9 )(x -2 ) = 0
⇒x=9 or x=2 Now V " ( x ) = 24 x - 132
V " (9 )= 2 1 6 - 132 = 84>0 
and V" (2) = 4 8 - 1 3 2 = —84 < 0 Thus, volume is maximum when x = 2 inch.
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A rectangular box is to be made from a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sidesWhat is the value of x for width the vulumeis maximum ?a)1 inchb)1.5 inchc)2 inchd)2.5 inchCorrect answer is option 'C'. Can you explain this answer?
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