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 1

The polynomial representing the amount of paper wasted when cutting the largest possible circle out of a square of side length l is (4-π)/4 * l².

Step-by-step explanation:

The question at hand is concerned with finding the polynomial that represents the amount of paper wasted when cutting out the largest possible circles from square pieces of paper. The side length of each square is given as l. The area of each square is l², while the area of the circle that can be cut from the square is calculated using the formula πr², where r is the radius of the circle. Because the largest circle that fits in the square touches all four sides, the diameter of the circle equals the side length l, making the radius r equal to l/2.

To find the polynomial for the wasted paper, first we calculate the area of the circle: π(l/2)², which simplifies to πl²/4. To find the wasted area, subtract the area of the circle from the area of the square: l² - πl²/4. This difference represents the wasted paper and can be further simplified to a single polynomial: (4/4)l² - (π/4)l², which simplifies to (4-π)/4 * l². This is the polynomial representing the amount of paper wasted for each square piece of paper.

Answer 2

Since the length of the square is l, the area would be l*l = l².
The area of the circle is given by the formula A=πr². In this case our radius, r, is 1/2 the length of the square, or 1/2l. This makes the area of the circle A=π(1/2l)²=π(1/4l²)=π(1/4)(l²)=1/4πl². (Remember, multiplication is commutative, which means we can switch these around to make them easier for us.)
The wasted area would be the area of the square minus the used area of the circle, or
l²-1/4πl²