A cashier is unsure whether a group of customers ordered and ate a large or a medium pizza. Medium pizzas have a diameter of 12 inches and large pizzas have a diameter of 16 inches. All that remains is one crust that has an arc length of 4.25 inches. All pizzas are divided into 12 equal slices. What size did the customers order? Explain how you determined your answer.
To determine the size of the pizza the customers ordered, we need to analyze the given information and perform some calculations.
First, let's calculate the circumference of the remaining pizza crust with an arc length of 4.25 inches. The circumference of a circle is given by the formula C = πd, where C represents the circumference and d represents the diameter.
For the remaining pizza crust, the arc length represents a portion of the circumference. Since the crust is a curved edge, we can assume it is an arc of a circle. To calculate the circumference, we need to find the proportion of the circumference that the given arc length represents.
To do this, we use the formula for the proportion of an angle to a full circle:
Proportion of arc length = (Angle of the arc / 360 degrees)
Given that there are 12 slices in total, each slice will have an angle of (360 degrees / 12 slices), which is equal to 30 degrees.
Using the formula for the proportion of an angle to a full circle, we can calculate the proportion of the arc length:
Proportion of arc length = (Angle of the arc / 360 degrees) = (30 degrees / 360 degrees) = 1/12
So, the arc length of 4.25 inches represents 1/12th of the circumference.
To calculate the circumference, we can use the formula:
Circumference = Arc length / Proportion of arc length
Circumference = 4.25 inches / (1/12) = 4.25 inches * 12/1 = 51 inches
Now that we know the circumference of the remaining pizza crust is 51 inches, we can determine the diameter of the pizza.
Circumference = πd
51 inches = πd
To solve for d (the diameter), we divide both sides by π:
d = 51 inches / π
Using the value of π as approximately 3.14, we can calculate the diameter:
d ≈ 51 inches / 3.14 ≈ 16.24 inches
Since the diameter is approximately 16.24 inches, it is closest to the diameter of a large pizza, which is 16 inches. Therefore, the customers most likely ordered a large pizza.
perimeter of 12 inch pizza = πd = 12π
so each of 12 pieces has a arc length of 12π/12 = π
perimenter of 16 inch pizza = 16π
so each of its 12 pieces has an arc length of 16π/12 = (4/3)π
π = 3.14159
(4/3)π = 4.189
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