Answer:
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
