I have a pizza. The radius is 10 inches long. The pizza was cut into 16 equal slices. When 1 slice was left, my sister and I both wanted it, so we agreed to cut it in half, but I like the crust more than she does, so we decided to cut it the "other way." In other words, the two pieces would not be symmetrical. The inside piece would contain all topping, and the outer piece would contain some topping and some crust.
1. Find the area of the whole pizza.
a=(pi)(10^2), =314
2. What is the area of one piece of pizza?
(pi)(r^2)/16, 100(pi)/16=19.6
3. What is the area of a half-piece?
100(pi)/36=10
4. What would the area of the whole pizza be if it were made of half pieces?
a=10(pi), a=100/10, a=10
5. What is the radius of a half-piece? (ie, where do I need to cut to make two equal halves out of a piece?)
I was wondering if I got these answers correct. If I did not, could someone please explain what I did wrong?Also, I was a little confused as to how to get the answer to the last question. Cheers!
I agree with 1. and 2.
In 3.
why are you dividing by 36.
If by the "area of a half-piece" you mean half the area of one of the 16 slices, then it would be
100π/32
I do not understand question #4
5. I will assume that your cut will be an arc.
so let the radius of that small piece be r
so πr^2 = (1/2)(19.6)
r = √(9.82/π) = 1.77 inches
If your cut is a straight line such that the smaller piece is an isosceles triangle, then the question would be a bit more difficult.
My answer to #5 is wrong,
go with bobpursley's way
in #5
I will assume that your cut will be an arc.
so let the radius of that small piece be r
so πr^2/16 = (1/2)(19.6)
r = 7.1 inches
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 (πr^2)/A = (πr^2)/(100π) = 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 (πr^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!