Example 31

A box is to be made from a sheet 12 ´ 12 sq. cm. by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut of, in order to have a box of the largest possible volume. Also find the largest volume of the box.

Solution : Let x cm be the length of the side of the square to be cut off.

Offer cutting of such four squares from four corners of the sheet 12 ´ 12 sq. cm.

The dimensions of the box to be made are length (L)

= (12 - 2x) cm, breadth (B)

= (12 - 2x) cm and height (H)

= x cm

Now volume of the box (v) = L.B.H

= (12 - 2x) (12 - 2x) x

\ v =

= (144 - 48x + 4x²) x

\ v = 4x³ - 48x² + 144x

\ = 12x² - 96x + 144

\ = 12 (x² - 8x + 12)

\ = 0

Þ (x² - 8x + 12) = 0

or (x - 6) (x - 2) = 0

OR x = 6 or x = 2 are critical points of which only x = 2 is

acceptable (why ?)

Now = 24x - 96

\ = 24 (2) - 96

= - 48 < 0

\ The volume of the box will be maximum when x = 2.

Therefore the square of side 2cm long must be cut off from each of the corner to get a box of maximum volume. Also L = (12 - 4) 8 cm, and H = 2 cm.

V_max = 8 ´ 8 ´ 2

= 128 sq. cm.

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Index

5.1 Tangent And Normal Lines
5.2 Angle Between Two Curves
5.3 Interpretation Of The Sign Of The Derivative
5.4 Locality Increasing Or Decreasing Functions 5.5 Critical Points
5.6 Turning Points
5.7 Extreme Value Theorem
5.8 The Mean-value Theorem
5.9 First Derivative Test For Local Extrema
5.10 Second Derivative Test For Local Extrema
5.11 Stationary Points
5.12 Concavity And Points Of Inflection
5.13 Rate Measure (distance, Velocity And Acceleration)
5.14 Related Rates
5.15 Differentials : Errors And Approximation

Chapter 6