My problem is this:
Three friends want to share a circular 14-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 7 inches.
My professor wants a neat and detailed solution but I don't know how to even begin to go about solving this, and things that I have seen online haven't helped me in the slightest
look at half the center 3rd
it is a sector of the circle, with right triangles on either side
the area of the sector is
... π * r^2 * (Θ/360)
... Θ being the central angle
the combined area of the two triangles is ... r sin(Θ/2) * r cos(Θ/2) = r^2 tan(Θ/2)
π r^2 / 6 = r^2 [tan(Θ/2) + (π Θ / 360)]
π / 6 = tan(Θ/2) + (π Θ / 360)
solve for Θ
the distance from the center to the edge of the middle 3rd is
... r sin(Θ/2)
To solve this problem, we need to find the distance from the center of the pizza where the two parallel cuts should be made. Here's a step-by-step solution:
1. Draw a circle representing the pizza with a radius of 7 inches.
2. Mark the center of the circle. This point represents the center of the pizza.
3. Draw two parallel lines across the diameter of the circle, dividing it into three equal sections.
4. Let's assume the distance from the center to the first cut is x. Since the pizza is divided into three equal sections, the distance from the second cut to the center will also be x.
5. Now, we have two parts of the pizza with equal size, along with the center part between the two cuts.
6. The circumference of the whole pizza is given by the formula C = 2πr, where r is the radius. In this case, C = 2π(7) = 14π inches.
7. The total length of the two outer pieces is equal to the circumference of the inner circle formed by the two cuts. This inner circle has a radius of 7 - x.
8. The circumference of the inner circle is C_inner = 2π(7 - x).
9. Since the two outer pieces and the middle piece should all have the same length, we can set up an equation:
C_outer + C_outer + C_inner = C
2C_outer + C_inner = C
2(14π - C_inner) + C_inner = 14π
28π - 2C_inner + C_inner = 14π
C_inner = 14π
10. Now, we have an equation for the circumference of the inner circle:
2π(7 - x) = 14π
7 - x = 7
x = 0
11. The distance from the center of the pizza where the cuts should be made is 0 inches.
Therefore, the two parallel cuts should be made at a distance of 0 inches from the center of the pizza.