Question

Mrs. Isabelle is making paper and plastic foam animals for her first-grade class. She is calculating the amount of wasted materials for environmental and financial reasons. Mrs. Isabelle is cutting circles out of square pieces of paper to make paper animals in her class. Enter the polynomial that represents the amount of paper wasted if the class cuts out the biggest circles possible in squares of length l. The polynomial that represents the amount of paper wasted is

Answer fast

Answer 1

We are given square with side length l. The largest circle that we can cut from that square has diameter equal to that side. This circle is called inscribed circle.

When cutting circle from square paper we will be left with some extra paper. Area of that extra paper can be calculated using formula:
area_of_extra_paper = area_of_square - area_of_circle

Now we can insert formulas:
$A = l^{2} - ( \frac{l}{2} )^{2} * \pi \\ A=l^{2} - \frac{l^{2}}{4} * \pi \\ A=l^{2} (1- \frac{ \pi }{4} )$

This is area of extra paper when we cut one circle. When we cut multiple circles we have:
$A=N*l^{2} (1- \frac{ \pi }{4} )$
Where:
N = number of circles
This formula is roughly:
$A=0.785*N*l^{2}$