A poster is to contain 72 cm2 of printed matter with border of 4cm each at the top and bottom and 2cm on each side .find the dimensions if the total area of the poster is minimum

Question

Sameer Chotrani · Answer

Let the length and width of the poster excluding the margins be x and y respectively.

According to the question,

xy=72, i.e., y=72/x …............(1)

Total length of poster (l) = 4+x+4 = (x+8) cm

Total width of poster (b) = 2+y+2 = (y+4) cm

Total area of poster (A) = lb

i.e., A = (x+8)(y+4)

i.e., A(x) = (x + 8)(72/x + 4) [from (1)]

i.e., A(x) = 72 + 4x + 592/x +32

i.e., A(x) = 4x + 576/x + 104

i.e., A’(x) = 4 – 576/x² + 0

i.e., A’(x) = (4x² – 576)/x² ….................(2)

At critical points,

A’(x) = 0

i.e., (4x² – 576) = 0 [from (2)]

i.e, x² = 144

i.e., x = 12 [ignoring negative square root since edge length can’t be nagative]

Putting in (1):

y = 72/12 = 6

To prove that this is a point of local minima you can use the 2^nd derivative test as follows:

A’’(x) = 1152/x³ [from using the equation above (2)]

i.e., A’’(12) = 1152/(12)³ > 0

This shows that local minima exists at x= 12.

Therefore, required dimensions are

Length (l) = x+8 = 12+8 = 20cm

Breadth (b) = y+4 = 6+4 = 10cm

Thank you for registering.

Register Now and Win Upto 25% Scholorship for a Full Academic Year !

Enter Your Details

Registration done!

A poster is to contain 72 cm2 of printed matter with border of 4cm each at the top and bottom and 2cm on each side .find the dimensions if the total area of the poster is minimum

A poster is to contain 72 cm2 of printed matter with border of 4cm each at the top and bottom and 2cm on each side .find the dimensions if the total area of the poster is minimum

1 Answers

Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy

Other Related Questions on Differential Calculus

ASK QUESTION

Answer and earn gift vouchers

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Complete Self Study Package designed by Industry Leading Experts

Live 1-1 coding classes to unleash the Creator in your Child

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Register Now & Win Upto 25% Scholorship