Question

A box is made with square base and open top. The area of the material used is 192 sq. cms. Find the dimensions of the box if its volume is maximum –

• A. 4, 4, 4
• B. 8, 8, 4
• C. 3, 4, 4
• D. 8, 4, 8

Answer 1

Answer: (B)

8, 8, 4

Answer 2

Let the length of each base side = x, height = H

\ Area of material used = area of base + area of 4 vertical sides

̃ x² + 4xH = 192 (given)

\ H = $\frac{192 - x^{2}}{4x}$ … (i)

Hence, the volume = ƒ(x) = x²H

= x²$\left\lbrack \frac{192 - x^{2}}{4x} \right\rbrack$ {using (i)}

\ ƒ(x) = $\frac{192x - x^{3}}{4}$

ƒ¢(x) = $\frac{1}{4}$ (192 – 3x²) = $\frac{3}{4}$ (64 – x²) = $\frac{3}{4}$

(8 – x) (8 + x)

For either maximum or minimum

Ģ(x) = 0 if it exists

Ģ(x) = 0

̃ x = 8, –8

But x ¹ –8, we consider x = 8 only

ƒ¢¢(x) = $\frac{1}{4}$ (0 – 6x)

ƒ¢¢(8) = $\frac{- 48}{4}$ = (–12) < 0

̃ ƒ has a maximum at x = 8

So that maximum volume is attained for x = 8

H = $\frac{192 - x^{2}}{4x}$ = $\frac{192 - 64}{32}$ = 4

Hence for maximum volume the dimensions are 8, 8, 4.