A rope of length 30cm is to be cut into two pieces

Question

A rope is to be cut into two piece ( x and y lengths). X is rolled into circle. Y is made into a square.

find a function (of x) for the combined area of the two figures.

Answer 1

The rope of length x will form the circumference of a circle, while the rope of length y=30-x will form the perimeter of a square. If the circle has radius r, we can use that the circumference of a circle is 2πr=x, which gives that r=x/(2π). Since the area of the circle is πr², we have that the area of the circle is equal to π(x/(2π))²=x²/(4π).

If the square (whose perimeter is y=30-x) has side length s, we can use that the perimeter of a square is 4s=30-x, which gives that s=(30-x)/4. Since the area of the square is s², we have that the area of the square is equal to [(30-x)/4]²=(30-x)²/16.

Thus the function f(x) = x²/(4π) + (30-x)²/16.

Answer 2

If the rope is 30 cm, then we will let x be one length, so x must be the other, and it will be 30 - x or y = 30 - x

x is to be a circle, so the circumference of the circle will give us the amount of rope used.

x = 2πr

solving for "r", this becomes r = x / (2π)

y is made into a square, so 4 L (where L is the length of one side of the square) must = y or

y = 4 L or the length of each side of the square is y/4 or L = y/4 = (30 - x)/4

The area of the circle is πr^2 and the area of the square is L^2, we need to add these together, substituting the bold expressions for "r" and for "L".

A = π [ x/(2π) ]^2 + [ (30-x)/4 ]^2

This is a function of x for the combined area, but your teacher will probably want it simplified.

A = π [ x^2 ] [ 1 / (4 π^2) ] + [ 900 - 60x + x^2 ] / 16

takes 5 or 6 more steps (that I am leaving to you) to find the most simplified function of

A(x) = [ 1/4 ] ( [ (4+π) / (4π) ] x^2 - 15x + 225 )