Mathematical Modelling Question

Question

A string 72 cm is to be cut into two pieces. One piece is used to form a circle and the other a square. What should be the perimeter of the square in order to minimize the sum of two areas?

Answer 1

Let x = perimeter of square, y = circumference of circle , r = radius of circle

circumference of circle = y = 2*pi*r

Given x + y = 72

then substituting for the circumference of circle

x + 2*pi*r = 72

solving for radius of circle in terms of x

r = (72 - x)/2*pi

area of square = x²/ 4

area of circle = pi*r²

Sum of the areas, substituting for r in terms of x:

x²/ 4 + 2*pi*[(72 - x)/2*pi]² =

Expanding

x²/ 4 + pi[(72)² - 2(72)x + x²]/4pi²] = x²/ 4 + (584 - 144x + x²)/4*pi =

Multiplying both terms by 4*pi and combining x²/4 + x^2:

5*pi*x²/ 4 - 144x + x² = 3.927x² - 144x + 5184

This a quadratic (in the form ax² + bx + c) which represents an upward-facing parabola when you graph it (can tell because coefficient of the x² term is positive).

Therefore to find the minimum x, you would find the x-term of the vertex of the parabola

(ex. the bottom of the 'U') using the vertex formula x = -b/2a.

b = -144, a = 5*pi*x²/ 4 = 3.927

x = 144/[2(3.927)] = 18.3346 which is the minimum perimeter for the square