Question

6. You are given a sheet of cardboard that is 8.0 in. x 11.0 in. How many full circles with a diameter of 3.0 inches can you cut from the cardboard? What is the percentage of waste if the

Answer 1

You can cut 12 full circles with a diameter of 3 inches from an 8 by 11 inch sheet of cardboard. The percentage of waste material from the cardboard is approximately 3.6%.

Step-by-step explanation:

You are given a sheet of cardboard that is 8.0 inches by 11.0 inches and you want to know how many full circles with a diameter of 3.0 inches you can cut from the cardboard, as well as the percentage of waste material.

First, calculate the area of the cardboard: Area = length × width = 8.0 in × 11.0 in = 88.0 in².

Then, calculate the area of a single circle: Area of circle = π × r², where r is the radius, which is half of the diameter. Therefore, r = 1.5 in. So, the area of one circle = π × (1.5 in)² ≈ 7.069 in².

Now, divide the area of the cardboard by the area of one circle to find out how many circles fit on the cardboard: 88.0 in² ÷ 7.069 in² ≈ 12.44. Since we can only cut full circles, we can cut 12 full circles.

To calculate the percentage of waste, we need to consider the area that is used by the circles compared to the total area. The used area is 12 circles × 7.069 in²/circle = 84.828 in². The waste is the total area minus the used area: 88.0 in² - 84.828 in² = 3.172 in².

The percentage of waste is (waste/total area) × 100 = (3.172 in² / 88.0 in²) × 100 ≈ 3.6%.

Answer 2

In order to answer this, it would be useful to calculate the area of the cardboard and the circles. We do as follows:

Total area of cardboard = 8.0 in. x 11.0 in = 88 in^2

Area of one circle = pi(r^2) = pi(0.5)^2 = 0.7854 in^2

Number of full circles = 88 / 0.7854 = 112.0451

Therefore, there will be 112 number of full circles. The percentage of waste is equal to:

% waste = (88 - 0.7854(112) / 88) x 100= 0.04%