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Example 28

A page is to contain 54 sq.cms of printed material. If the margins are 1cm at top and bottom and 0.5cm at sides, find the most economical dimensions of the page.

Solution : Let the dimension of the printed

material be x cm. and y cm.

Then xy = 54 \ y =

The dimension of the page are

(x + 3) and (y + 2) cm. Then the area

of the page is A = (x + 3) (y + 2)

\ A = (x + 3)

= 60 + + 2x

Then = and critical points are x = + 9

Also, =

\

\ The relative minimum is at x = 9

\ The required dimensions of the page are (x + 3) = 12 cm and y = 54 /9 = 6 Þ (y + 2) = 6 + 2 = 8 cm i.e. 12 cm wide and 8 cm high.

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Example 29

A 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?

Solution : Let the dimensions of the base of the open box be x cm and x cm and height be y cm.

\The area of the material of the open box = (area of the base) + 4 (area of vertical face)

\ 192 = (x ´ x) + 4 (xy)

\ 192 = x2 + 4xy

\ y =

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