A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Explanation

`V(x)= x(18-2 x)^{2} `
`therefore V^{prime}(x) =(18-2 x)^{2}-4 x(18-2 x) `
` =(18-2 x)[18-2 x-4 x] =(18-2 x)(18-6 x) `
` = 6 times 2(9-x)(3-x) = 12(9-x)(3-x) `
` { And, } V^{prime prime}(x)=12[-(9-x)-(3-x)] =-12(9-x+3-x) `
` =-12(12-2 x) =-24(6-x) `
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Now, V^{prime}(x)=0 Rightarrow {x}=9 or {x}=3`

If` x=9, `then the length and the breadth will become `0 . x!= 9. `
`

Rightarrow x=3 . `
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Now, quad V^{prime prime}(3)=-24(6-3)=-72<0`