Page 161 - C:\Users\trainee\AppData\Local\Temp\msoEAA3.tmp

Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
Differentiating equation (1.85) with respect to (x, y and λ), we get:

∂L
= y + 2λ = 0 = y = −2λ - - - - (1.86)
∂x

∂L
= x + 2λ = 0 = x = −2λ - - - - (1.87)
∂y
∂L
= 2x + 2y − 12 = 0 = x + y = 6 - - - - (1.88)
∂λ

Substituting the values of ‘x’ and ‘y’ in equation (1.88), we get:
3
−2λ − 2λ = 6 = −4λ = 6 or λ = −
2

Substituting the value of ‘λ’ in equations (1.86) and (1.87) we get the values of ‘x’ and ‘y’ to be 3.

The optimum area of the rectangle is got by substituting the value of (x, y and λ) in equation
(1.85):

3
L = 3 ∗ 3 − 6 + 6 − 12 = 9
2

It can be observed that, when the sum of ‘n’ numbers is given, each number is given by (sum/n)
for the product to be maximum.

Š‹• ‹• ƒ‘–Š‡” ‘’–‹‹œƒ–‹‘ –‡ Š‹“—‡ ™Š‡”‡ ϐ‹†‹‰ –Š‡ ƒš‹— ƒ† ‹‹— ˜ƒŽ—‡•
‹• ‘•–”ƒ‹‡†Ǥ

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
Page 64