Let's go through each question to check if your answers are correct:
1. Find the area of the whole pizza.
You correctly used the formula for the area of a circle, which is A = πr^2. Substituting the radius value of 10 inches, the calculation should be A = π(10^2) = 100π ≈ 314.16 square inches. So, it seems there was a rounding error in your answer. The correct value rounded to two decimal places is 314.16 square inches.
2. What is the area of one piece of pizza?
To find the area of one slice, we divide the area of the whole pizza by the number of slices, which is 16 in this case. So, the calculation should be A = (100π)/16 ≈ 6.28 square inches. It seems you made an error with this calculation as well. The correct value rounded to two decimal places is 6.28 square inches.
3. What is the area of a half-piece?
To find the area of a half-piece, we divide the area of one slice by 2. So, the calculation should be A = (100π)/32 ≈ 3.14 square inches. You made an error in this calculation too. The correct value rounded to two decimal places is 3.14 square inches.
4. What would the area of the whole pizza be if it were made of half pieces?
If the pizza were made of half pieces, we would have twice the number of slices, which is 32. So, the calculation should be A = (100π)/32 ≈ 9.87 square inches. You made an error in this calculation as well. The correct value rounded to two decimal places is 9.87 square inches.
5. What is the radius of a half-piece?
To determine where to make the cut to create two equal halves out of a piece, you need to find the radius of the half-piece. Since the area of a circle is proportional to the square of its radius, the ratio of the areas of the half-piece and the whole pizza would be the same as the ratio of their radii squared.
Let's denote the radius of the half-piece as r. So, the calculation would be (Ï€r^2)/A = (Ï€r^2)/(100Ï€) = r^2/100. Since the area of a half-piece is one-third of a full slice (10 square inches), we can set up the equation (Ï€r^2)/2 = 10.
Simplifying the equation, we get (πr^2) = 20. Substituting π ≈ 3.14, the equation becomes 3.14r^2 = 20. Solving for r^2, we find r^2 ≈ 20/3.14 ≈ 6.37. Taking the square root of both sides, we get r ≈ 2.52 inches (rounded to two decimal places).
So, it seems your answer to the last question is incorrect. The correct radius of a half-piece would be approximately 2.52 inches.
I hope this clarifies things for you!