A wire of length 18 m is used to form a circle and a square. Find the radius of the circle and the side length of the square that produce a minimum total area?

1 Answer
Jul 10, 2018

Answer:

  • # {(r approx 1.26m ),(s approx 2.52m ):}#

Explanation:

If the wire of length is split into 2 lengths:

  • #l_1, l_2 qquad " with "l_1 + l_2 = 18 #

... to make circle and square respectively.

Circle

Circle will have circumference #C#:

  • #C = 2 pi r = l_1 implies r = l_1/(2 pi )#

  • #implies A_1 = pi r^2 = l_1^2/(4 pi )#

Square

Square will have perimeter #p# with side #s#:

  • #p = 4 s = l_2 implies s = l_2/4 #

  • #implies A_2 = s^2 = l_2^2/16#

Total Area

#A(bbl) = A_1(l_1) + A_2(l_2) = l_1^2/(4 pi ) + l_2^2/16 #

#= l_1^2/(4 pi ) + (18 - l_1)^2/16 #

This is now in single variable. Differentiate wrt #l_1# to optimise:

#A' = l_1/(2 pi) - (18 - l_1) /8 = 0 #

#implies {(l_1 = (18 pi)/(4 + pi) ),(l_2 = 72/(4 + pi) ):} qquad {(r approx 1.26m ),(s approx 2.52m ):}#

Using approximate values for #l_1,l_2#:

#A approx 11.3 \ m^2#

The 2nd derivative is:

#A'' = 1/(2 pi)+ 1/8 > 0 #

So this is a minimum .