Calculus Question (urgent help needed)

Question

A small piece of string that measures 20cm in length is cut into two pieces. The first is made into a circle with radius 'r ' in (cm) and the second a square with side length 's' in (cm).

 

Write down an expression that gives the sum of the Areas 'A' enclosed by these two shapes in terms of 'r' Then use differentiation to determine the value of 'r' for which 'A' is a minimum.

 

## So far i have this; Perimeters of both shapes (20cm) = 2(pi)r + 4s, by making 's' subject we get s= (20-2(pi)r)/2.

Therefore area 'A' = (pi)r^2 + s^2,  if we sub 's' in we get - A = (pi)r^2 + ((20-2(pi)r)/2)^2

 

Now here is where i get stuck assuming where i have got to is correct what do i do next? Any help would be great :-)

 

Shannon xx

Answer 1

Hi Shannon,

I like the way you set up the problem, but I computed a different value for s in terms of r. (π) means pi

Perimeter = 20 = 2πr + 4s so s=(20-2πr)/4 = (5-0.5πr)

Then A = πr² + (5-0.5πr)² = πr² + 25 - 5πr + 0.25π²r²

setting the derivative = 0 to find the minimum area gives

0=2πr -5π + 0.5π²r

Dividing both sides by pi gives 0=2r -5 +0.5πr

so 5=r(2+0.5π) and r = 1.400cm

I hope this helps.