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  • 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, the

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 cm^{^{2}}) is equal to :

Option: 1

800


Option: 2

1025


Option: 3

900


Option: 4

675


Answers (1)

best_answer

Let the side of the square to be cut off be x cm.
Then, the length and breadth of the box will be (30 – 2x) cm each and the height of the box is x cm therefore

The volume V(x) of the box is given by

\begin{aligned} & V(x)=x(30-2 x)^2 \\ & \frac{d v}{d x}=(30-2 x)^2+2 x \times(30-2 x)(-2) \\ & 0=(30-2 x)^2-4 x(30-2 x) \\ & 0=(30-2 x)[(30-2 x)-4 x] \\ & 0=(30-2 x)(30-6 x) \\ & x=15,5 \end{aligned}

x \neq 15        (Not possible)

\left \{ \therefore V=0 \right \}

Surface area without top of the box   = lb+2(bh+hl)

\begin{aligned} & =(30-2 x)(30-2 x)+2[(30-2 x) x+(30-2 x) x] \\ & =\left[(30-2 x)^2+4\{(30-2 x) x\}\right. \\ & =\left[(30-10)^2+4(5)(30-10)\right] \\ & =400+400 \\ & =800 \mathrm{~cm}^2 \end{aligned}

 

 

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