Question

To make circular place mats, a company cuts circles out of plastic squares. The circles are cut as wide as the squares to lessen the amount of wasted material. Use the drop-down menus below to complete statements about the amount of wasted material for circular place mats with a diameter of d.

Answer 1

To determine the amount of wasted material for circular place mats with a diameter of d cut from squares, calculate the area of the square as d² and the area of the circle as π(d/2)². The difference, d² - π(d/2)², is the wasted material.

Step-by-step explanation:

Understanding the Waste Material in Circular Place Mats

When a company cuts out circular place mats from plastic squares to minimize waste, it seeks to maximize the usage of material. For a place mat with a diameter d, the square from which it is cut will have sides of length d as well. The area of such a square will be d times d, or d². However, the area of the circular place mat, which has a radius of d/2, will be π times the radius squared, or π(d/2)². The wasted material would be the difference between the area of the square and the area of the circle, which is d² - π(d/2)².

To visualize the difference between the areas, if the diameter of the circle, d, is 2 meters, then the area of the circular place mat would be less than the area of the square, which would be 4 square meters. More precisely, the area of the circle would be π times 1 square meter, since the radius is half the diameter, in this case, 1 meter. Thus, the area of the circle is approximately 3.14 square meters, and the wasted area would be approximately 0.86 square meters (4m² - 3.14m²).