The box with a square base is to have an open top. The area of the material for making the box is $192 s q . c m$ . What should be its dimensions in order that the volume is as large as possible?

Open in App

Solution

$\begin{matrix} L e t t h e d e n s i o n s o f t h e b a s e o f t h e o p e n b o x b e x c m a n d x c m a n d h e i g h t b e y c m T h e a r e a o f t h e m a t e r i a l o f t h e o p e n b o x \Rightarrow (A r e a o f t h e b a s e) + 4 (a r e a o f v e r t i c a l f a c e) \Rightarrow 192 = (x \times x) + 4 (x y) \Rightarrow 192 = x^{2} + 4 x y \Rightarrow \frac{192 - x^{2}}{4 x} = y L e t v b e t h e v o l u m e o f t h e b o x, V = b a s e a r e a \times h e i g h t = x^{2} y = x^{2} (\frac{192 - x^{2}}{4 x}) ∴ V = x (\frac{192 - x^{2}}{4}) = \frac{192 x - x^{3}}{4} \frac{d v}{d x} = \frac{192 - 3 \times x^{2}}{4} \frac{d v}{d x} = 0 x^{2} = 64 sin c e, x i s l e n g t h o f t h e s i d e, i t c a n n o t b e 0 - v e ∴ x \neq - 8 a n d, \frac{d^{2} v}{d x^{2}} = \frac{- 3}{2} x {(\frac{d^{2} v}{d x^{2}})}_{x = 8} = \frac{- 3}{2} (8) V = i s max i m u m w h e n x = 8 c m w h e n x = 8, t h e n y = \frac{192 - 8^{2}}{4 (8)} = 4 c m ∴ D i m e n s i o n s a r e 8 c m & 4 c m l e s t p o s s i b l e v o l u m e o f b o x (c u b e) V = x^{2} y = {(8)}^{2} (4) = 256 s q c m \end{matrix}$

Suggest Corrections

Similar questions

Q. A box is made with square base and open top. The area of the material used is 192 sq. cms. If the volume of the box is maximum, the dimensions of the box are

Q. A closed (cuboid) box with a square base is to have a volume

2000 c . c

. The material for the top and bottom of the box is to cost Rs.

3

per square

c m

and the material for the sides is to cost Rs.

1.50

per square

c m

. If the cost of the materials is to be the least, find the dimensions of the box.

Q. A rectangular box with an open top is constructed from cardboard to have a square base of area

x^{2}

and height

h

. If the volume of this box is

50

cubic units, determine how many square units of cardboard are required to make this box ( in terms of

x

A metallic sheet of rectangular shape with dimensions 48mx36m. From each of its corners a square is cut off so as to make an open box. If the length of the square is 8m, the volume of the box is

Q. A closed rectangular box with a square base is to be made so as to contain

1000

cubic feet. The cost of the material per sq. foot for the bottom is

15

paisa, for the top

254

paisa, and for the sides

20

paisa. The labour charges for making the box are Rs.

3

. Find the dimensions of the box when the cost is minimum.

Join BYJU'S Learning Program

The box with a square base is to have an open top. The area of the material for making the box is 192 sq.cm. What should be its dimensions in order that the volume is as large as possible?

The box with a square base is to have an open top. The area of the material for making the box is $192 s q . c m$ . What should be its dimensions in order that the volume is as large as possible?