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's class is making plastic foam spheres out of plastic foam cubes. Enter the polynomial that
represents the amount of plastic foam wasted if the class cuts out the biggest spheres possible from
cubes with side lengths of /. The volume of a sphere of radius r is 1³.
3
The polynomial that represents the amount of plastic foam wasted if the class cuts out the biggest
spheres possible from cubes with side lengths of/ is

Answer 1

The polynomial that represents the amount of foam wasted is W_{wasted} = (1 - (\pi/6))l^3, which is found by subtracting the volume of the largest possible sphere from the volume of the cube with side length l.

Step-by-step explanation:

The question asks to define the polynomial that represents the amount of plastic foam wasted when making the biggest sphere possible from a cube with side length l. The volume of a cube is given by Vcube = l3, and the volume of the largest sphere that can be fitted inside the cube is given by Vsphere = 4/3πr3, where the radius r of the sphere is equal to half the side length of the cube, i.e., r = l/2.

Substituting r = l/2 into the formula for the volume of the sphere, we get Vsphere = 4/3π(l/2)3. Simplifying, Vsphere = (4/3πl3)/8. To find the wasted material, subtract the volume of the sphere from the volume of the cube to get the polynomial Wwasted = l3 - (4/3πl3)/8. Simplifying, the polynomial is Wwasted = (1 - (π/6))l3.