Question

A rectangular sheet of metal is 1.2 meters long and 80 centimeters wide. From this sheet, a machine cuts metal circuits. a. Find the maximum number of circles with a diameter of 4 cm that can be cut from sheet metal. b. Eli says less than 25% of the foil will go to waste. Is Eli right? Justify your answer. * c. The 4 cm diameter of the circles is given rounded to the nearest millimeter. How would this affect responses to request a and b?

Answer 1

Diameter of the circle = 4 cm

So area of each circle = $\pi r^2=4\pi \ cm^2$

and area of rectangular sheet = $80\ cm\times 120\ cm=9600\ cm^2$

(a) Along length of rectangular sheet

Maximum number of circles that can fit = $\dfrac{120\ cm}{4\ cm}=30$

Along breadth of the rectangular sheet

Maximum number of circles that can fit = $\dfrac{80\ cm}{4\ cm}=20\ circles$

So,

Maximum number of circles with diameter of 4 cm that can be cut from sheet metal = $30\times 20=600$

(b) Area of 600 circles = $600\times 4\pi=2400\pi=7539.82\ cm^2$

So area of rectangular foil that is wasted = $9600-7539.82=2060.17\ cm^2$

So, percentage of area that is wasted = $\dfrac{2060.17}{9600}\times 100=21.46\ \%$

Yes, Eli is right that less than 25% of area is wasted.

(c) For minimum area to be wasted, the circles should be perfectly 4 cm in diameter

If diameter of circles is greater than 4cm let's say 4.01 cm then

maximum number of circles that can be formed = 29 x 19=551

So, more area will be wasted.