Question

A wire of length 36cm is to be cut into 2 piece one of the piece is made a circle and other into an equivalent triangle what should be length of the two piece so that the combined area of both circle an equivalent triangle is minimum

Answer 1

econd Derivative Test - If f'(x) = 0 at a point and f''(x) > 0 at this point, then f(x) has a local minimum at this point.

 

Let s be the side of square and a be side of equilateral triangle.

 

Given,

 

4s+3a=36

a=36−4s3  -- ( 1 )

 

Total area A=s2+3√4a2

Substituting for a from (1) above,

 

A=s2+3√4(36−4s3)2

A=s2+43√9(9−s)2

 

dAds=2s+43√9(2s−18)

d2Ads2=2+83√9>0

 

Hence based on second derivative test, a minimum happens for A wherever dAds=0

 

So,

 

dAds=2s+43√9(2s−18)=0

 

s=83√2+83√9=363√9+43√

 

Substituting in ( 1 ) and simplifying, we get

 

a=1089+43√

 

So size of two pieces are 4s and 3a with s and a with values as above.