When the waiter was asked to slice the 16-inch (in diameter) pizza into exactly 4 slices of equal size, he decided to do so using concentric circles rather than radial slices. Determine the radii of each of the three cuts he had to make.
By equal size I assume you mean equal area. The total pie area is 3.14(8^2) = 201.06 sq.in. Each piece must therefore be 50.265 sq.in. in area.
The central piece is a circle with area = 50.265 = 3.14(r1)^2 making r1 = 4 in.
The next piece is a circular ring of r2 outer radius and r1 inner radius and area = 50.265 = 3.14(r2^2 - r1^2) making r2 = 5.656 in.
The next piece is a circular ring of r3 outer radius and r2 inner radius and area = 50.265 = 3.14(r3^2 - r^2) making r3 = ? Your turn.
The person getting the outer ring is cheated out of some cheese.
To slice the 16-inch pizza into 4 equal-size slices using concentric circles, the waiter would need to make 3 cuts.
Let's start by finding the radius of the pizza. The diameter of the pizza is 16 inches, so the radius would be half of that.
Radius = Diameter / 2
Radius = 16 inches / 2
Radius = 8 inches
Now, the waiter needs to make three cuts that divide the pizza into four equal-size slices. Since the cuts are made using concentric circles, the radii of the cuts will be increasing in length.
First Cut: The first cut is made at the center of the pizza, dividing it into two equal halves. Therefore, the radius of the first cut is 0 inches.
Second Cut: The second cut needs to divide one of the halves into two equal parts. To do this, the radius of the second cut needs to be half the radius of the pizza.
Radius of the second cut = Radius of the pizza / 2
Radius of the second cut = 8 inches / 2
Radius of the second cut = 4 inches
Third Cut: The third cut needs to divide one of the remaining slices into two equal parts. So, the radius of the third cut should be half the radius of the previous cut.
Radius of the third cut = Radius of the second cut / 2
Radius of the third cut = 4 inches / 2
Radius of the third cut = 2 inches
In conclusion, the radii of each of the three cuts the waiter had to make to slice the 16-inch pizza into 4 equal-size slices using concentric circles are 0 inches, 4 inches, and 2 inches respectively.
To determine the radii of the cuts the waiter made to slice the pizza into 4 equal-sized slices using concentric circles, we can follow these steps:
Step 1: Calculate the radius of the pizza:
The diameter of the pizza is given as 16 inches. Recall that the radius is half of the diameter. So, the radius of the pizza is 16/2 = 8 inches.
Step 2: Divide the pizza into four slices:
Since the waiter decided to use concentric circles instead of radial slices, imagine that he created three concentric circles within the pizza.
Step 3: Determine the radii of the concentric circles:
The waiter wants to divide the pizza into four equal-sized slices. So, we need to find three radii that would create four concentric circles with equal areas.
Let's call the radii of the concentric circles r1, r2, and r3. The largest circle, with radius r1, encompasses the entire pizza. The second circle, with radius r2, is the boundary for the three slices, and the smallest circle, with radius r3, defines the innermost part of the slices.
Since the four slices should have equal areas, we know that the area of each slice is equal to one-fourth of the total area of the pizza.
Step 4: Calculate the areas of the circles:
The area of a circle can be calculated using the formula A = π * r^2, where A is the area and r is the radius.
For the largest circle, its radius r1 is equal to the radius of the whole pizza, which is 8 inches. Therefore, the area of the largest circle is A1 = π * (8^2) = 64π.
For the second circle, its area should be three times the area of each slice. So, its area A2 is equal to (1/4) * (total area of the pizza), which is (1/4) * (64π) = 16π.
For the smallest circle, its area A3 should be equal to the area of each slice. Therefore, A3 = (1/4) * (total area of the pizza), which is (1/4) * (64π) = 16π.
Step 5: Solve for the radii:
Now we need to find the radii r2 and r3. Since we know the areas of circles 2 and 3, we can use the formula for the area of a circle to solve for their radii.
For the second circle, A2 = π * (r2^2) = 16π.
Dividing both sides by π, we get: r2^2 = 16.
Taking the square root of both sides, we find: r2 = √16 = 4 inches.
For the third circle, A3 = π * (r3^2) = 16π.
Dividing both sides by π, we get: r3^2 = 16.
Taking the square root of both sides, we find: r3 = √16 = 4 inches.
Therefore, the waiter had to make three cuts using concentric circles with radii r1 = 8 inches, r2 = 4 inches, and r3 = 4 inches to slice the 16-inch pizza into four equal-sized slices.