Three friends want to share a circular 12-inch pizza equally by
exactly only two parallel cuts. How far from the center must the
cuts be? Hint: The pizza is a circle of radius 6 inches.
If the cuts are at a distance of x from the center, then since the area of a circular segment is
a = 1/2 r^2(θ-sinθ)
we want that to be 1/3 of the area, or π/3 r^2
So,
1/2 r^2(θ-sinθ) = π/3 r^2
3(θ-sinθ) = 2π
Solve that using your favorite method, and we have
θ = 2.6
So, x/r = cos θ/2
x = 1.6 in from the center
sadaf
To determine how far from the center the two parallel cuts must be, we need to divide the pizza into three equal parts. Here's how we can do it:
1. Start by drawing a circle with a radius of 6 inches, representing the pizza.
2. Divide the circle into three equal arcs by drawing two radii from the center. Each angle between the radii will be 120 degrees.
3. Now, we need to find the length of each arc corresponding to each friend's share of the pizza. Since we have a total angle of 360 degrees (the entire pizza), and we want to divide it into three equal parts, each friend's share will be 360/3 = 120 degrees.
4. To find the length of the arc, we need to calculate the circumference of the circle and divide it by the total angle. The formula for the circumference of a circle is C = 2 * π * r.
In this case, the radius (r) is 6 inches, so the circumference (C) = 2 * π * 6 = 12π inches.
The length of each arc that represents a friend's share will be (12π * 120/360) = 4π inches.
5. Since we want to divide the pizza into two equal parts using parallel cuts, we need to place the cuts at equal distances from the center.
The total distance between the parallel cuts will be 2 times the radius (2 * 6 = 12 inches).
Divide this total distance by 3 to find the distance from the center that each cut should be placed: 12/3 = 4 inches.
Therefore, the parallel cuts must be placed 4 inches from the center of the pizza.