Question

10 in.
6 in.
for a class project, a teacher cuts out 15 congruent circles from a single sheet of paper that measures 6 inches by 10 inches. how much paper is wasted?
\\( (60 - 15\pi) \\) square inches
\\( 15\pi \\) square inches
45 square inches
\\( (60 - \pi) \\) square inches

Explanation:

Step1: Find area of the paper

The paper is a rectangle with length \( 10 \) inches and width \( 6 \) inches. The area of a rectangle is \( \text{length} \times \text{width} \), so the area of the paper is \( 10\times6 = 60 \) square inches.

Step2: Find the radius of each circle

From the diagram, we can see that the height of the rectangle (6 inches) is equal to the diameter of 3 circles. So the diameter of one circle \( d=\frac{6}{3}=2 \) inches, then the radius \( r = \frac{d}{2}=1 \) inch.

Step3: Find the area of one circle

The area of a circle is \( \pi r^{2} \), with \( r = 1 \), so the area of one circle is \( \pi\times1^{2}=\pi \) square inches.

Step4: Find the total area of 15 circles

The total area of 15 congruent circles is \( 15\times\pi=15\pi \) square inches.

Step5: Find the wasted area

The wasted area is the area of the paper minus the total area of the circles, so it is \( 60 - 15\pi \) square inches.

Answer:

\( (60 - 15\pi) \) square inches