Three students order a pizza that they want to split evenly. They decide to cut it with 2 parallel lines. If pizza has a diameter 18 inches, at what point should they cut it to make sure the three parts are the same size?
Draw a circle of radius 9 centered at (0,0)
Work with just the top half. If it can be divided into 3 equal parts by two vertical lines, so will the whole circle.
In fact, just work with the upper-right quarter-circle. If it can be divided by one vertical line so that the left section is half as big as the right section, then by symmetry another line on the other side of the y-axis will do the job.
So, we want a line at x=a where
2∫[0,a]√(81-x^2) dx = ∫[a,9]√(81-x^2) dx
Now, ∫√(81-x^2) dx = 1/2(x√(81-x^2) + 81 arcsin(x/9))
2(a√(81-a^2)+ 81arcsin(a/9))-(0+0) = (0+81*π/2)-(a√(81-a^2)+81 arcsin(a/9))
3(a√(81-a^2) + 81arcsin(a/9)) - 81π/2 = 0
Graphing that reveals that a = 2.3844
So, two lines 2.3844 inches from the center will do the trick.
To cut the pizza into three equal parts using two parallel lines, we need to divide the pizza into three sections with the same area.
Here are the step-by-step instructions:
1. Find the radius of the pizza by dividing the diameter by 2:
radius = 18 inches / 2 = 9 inches.
2. Calculate the area of the whole pizza using the formula for the area of a circle:
area = π * radius^2
area = 3.14 * (9 inches)^2
= 3.14 * 81 square inches
= 254.34 square inches.
3. Divide the total area of the pizza by 3 to find the area of each section:
area of each section = total area / 3
= 254.34 square inches / 3
= 84.78 square inches.
4. Draw two parallel lines on the pizza, dividing it into three sections.
The lines should divide the pizza into equal areas.
Using a ruler or another straight edge, draw one line passing through the center of the pizza, dividing it into two equal halves.
Then draw a second line at an angle to the first line, intersecting it at the center of the pizza as well. This will divide the pizza into three equal sections.
Each section should have an area of approximately 84.78 square inches.
Remember to use a ruler or other tools when physically dividing the pizza to ensure precise and accurate cuts.
To determine where to cut the pizza, we need to find the point on the circumference where the two parallel lines should intersect.
First, let's imagine the pizza as a circle with a diameter of 18 inches. The circumference of a circle can be found using the formula C = πd, where C is the circumference and d is the diameter. In this case, the circumference would be 18π inches.
Since the pizza needs to be evenly split into three parts, we can divide the circumference by 3 to find the length of each part. Thus, each part should be (18π)/3 inches long.
To find the point on the circumference where the two parallel lines should intersect, we can divide the circumference of the pizza by 3, because there are three equal parts. This will give us the length of each part.
Therefore, the point where they should cut the pizza is (18π)/3 inches from the starting point.