Emily cuts two circles from a sheet of colored paper measuring 8" x 12". One circle had a radius of 3 inches and the other had a diameter of 5 inches. How many square

inches of paper are left over? Is it possible to cut another circle with a 3 inch radius from

the paper? If the 3 inch circle fits what is the largest circle (What is its radius) you can

draw that will still fit on your paper.

a. 48.1 square inches

b. yes

c. 2.51 in

Step-by-step explanation:

a. How many square  inches of paper are left over?

To know this, we need to calculate the area of the sheet of colored paper and then subtract the are of the circle from it.

So, A = area of colored paper = 8" x 12" = 96 in²

A'= Area of 3 inch radius circle = πr²

= π(3)²

= 9π in²

A" = Area of 5 inch diameter circle = πd²/4

= π(5)²/4

= 25π/4 in²

= 6.25π in²

A₀ = area of circles = 9π in² + 6.25π in²

= 15.25π in²

= 47.91 in²

So, there are left A₁ = A - A₀

= 96 in² -  47.91 in²

=  48.09 in²

≅ 48.1 in²

So there are 48.1 square inches left over.

b. Is it possible to cut another circle with a 3 inch radius from  the paper?

To know this, we calculate the area of a 3 inch radius paper and see if it is less or more than the remaining area of paper. If it is less, it can be cut.

So, A₂ = Area of 3 inch radius circle = πr²

= π(3)²

= 9π in²

= 28.3 in²

Since A₂ = 28.3 in² < A₁ = 48.1 in² (the rest of the colored paper area), it can be cut. So the answer is yes, another circle with a 3 inch radius can be cut from  the paper.

c. If the 3 inch circle fits what is the largest circle (What is its radius) you can  draw that will still fit on your paper.

We find the area of the remaining colored paper by subtracting the area of the 3 inch circle from the remaining area. So.

A₃ = A₁ - A₂

= 48.1 in² - 28.3 in²

= 19.8 in²

This is the area of the remaining colored paper. We then find the radius, r of the circle with area A₃ = 19.8 in² that would fit into the area.

So, A₃ = πr²

r = √(A₃/π)

= √(19.8 in²/π)

= √(6.302 in²)

= 2.51 in