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
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 = 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 = 4 s = l_2 implies s = l_2/4 # 
#implies A_2 = s^2 = l_2^2/16#
Total Area
This is now in single variable. Differentiate wrt
Using approximate values for
The 2nd derivative is:
So this is a minimum .