Robert made seven pepperoni pizzas for a party. He cut the pepperoni himself from a sausage that was 15 inches long and 1 1/4 inches in diameter. The pizzas were each 18 inches in diameter. His friend Hannah is a real math whiz (a math freak some say) and took the opportunity to calculate what percent of the total pizza surface area was covered by pepperoni. The pepperoni was cut into slices 1/8 of an inch thick. Each pizza has the same number of whole pepperoni slices. (He ate any extra pepperoni!) What percent of the total pizza surface area was covered by the pepperoni? Round your answer to the nearest hundredth.
First, let's calculate the total surface area of one pizza. The formula for the surface area of a circle is A = πr^2, where A is the area and r is the radius. Since the diameter of the pizza is 18 inches, the radius is half of that, which is 9 inches.
So the surface area of one pizza is A = π(9^2) = 81π.
Next, let's calculate the volume of the sausage that Robert used to make the pepperoni. The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height. The height of the sausage is its length, which is 15 inches, and the radius is half of its diameter, which is 1 1/4 inches / 2 = 5/8 inches.
So the volume of the sausage is V = π(5/8)^2 * 15 = 9.375π.
Since the sausage was sliced into 1/8-inch thick slices, we can calculate the number of slices by dividing the height of the sausage by the thickness of each slice. The height of the sausage is 15 inches and the thickness of each slice is 1/8 inch, so there are 15 / (1/8) = 15 * 8 = 120 slices.
Now, let's calculate the total surface area covered by the pepperoni. The surface area covered by one slice can be calculated using the formula for the area of a circle: A = πr^2, where A is the area and r is the radius. Since the diameter of each slice is the same as the thickness (1/8 inch), the radius is half of that, which is 1/16 inch.
So the surface area covered by one slice is A = π(1/16)^2 = π/256.
Therefore, the total surface area covered by the pepperoni is 120 * π/256 = 15π/32.
Finally, let's calculate the percentage of the total pizza surface area covered by the pepperoni. The total surface area of the seven pizzas is 7 * 81π = 567π.
So the percentage of the total pizza surface area covered by the pepperoni is (15π/32) / (567π) * 100% = (15/32) / 567 * 100% =~ 0.0791406 * 100% =~ 7.91%.
Therefore, approximately 7.91% of the total pizza surface area was covered by the pepperoni.
area of pizzas = 7*81π = 567π
# slices: 15*8 = 120
since all pizzas have the same number of slices, that would be [120/7] = 17 slices per pizza, with one left over
area of 119 slices: 119 * (5/8)^2 π = 46.48π
% of surface covered by pepperoni = 46.48/567 = 0.08198 or 8.2%
Apologies for the incorrect calculations. You are correct.
The area of the pizzas is indeed 567π.
Since there are 120 slices and 7 pizzas, each pizza has 120/7 = 17 slices approximately.
The area covered by 119 slices is 119 * (5/8)^2 π = 46.48π approximately.
The percentage of the total pizza surface area covered by the pepperoni is 46.48/567 * 100% = 0.08198 * 100% = 8.2% approximately.
Therefore, approximately 8.2% of the total pizza surface area was covered by the pepperoni.
To calculate the percent of the total pizza surface area covered by the pepperoni, we need to find the total surface area of the pizzas and the surface area covered by the pepperoni.
1. Calculate the total surface area of one pizza:
The surface area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius. Since the diameter of the pizza is given (18 inches), we can find the radius by dividing the diameter by 2.
Using this formula, the surface area of one pizza is:
A = π(9)^2 = 81π square inches.
2. Calculate the total surface area of all seven pizzas:
Since each pizza has the same surface area, we can multiply the surface area of one pizza by the number of pizzas:
Total surface area = 7 * 81π = 567π square inches.
3. Calculate the surface area covered by the pepperoni:
The length of the sausage is irrelevant; what matters is the total thickness of the slices. Since each slice is 1/8 of an inch thick, we need to multiply the thickness by the number of slices.
The number of slices can be found by dividing the length of the sausage by the thickness of one slice:
Number of slices = 15 inches / (1/8 inch) = 15 * 8 = 120 slices.
The surface area covered by one slice of pepperoni can be approximated by:
A = πr^2, where r is the radius of the slice.
Since the diameter of the sausage is 1 1/4 inches, the radius is 5/8 inches (1/2 of the diameter).
A = π(5/8)^2 = 25π/64 square inches.
To find the total surface area covered by the pepperoni, we multiply the area of one slice by the number of slices:
Total pepperoni surface area = (25π/64) * 120 = 75π/8 square inches.
4. Calculate the percentage of the total pizza surface area covered by the pepperoni:
To find the percentage, we divide the surface area covered by the pepperoni by the total surface area of the pizzas and multiply by 100:
Percentage = (Total pepperoni surface area / Total surface area) * 100
Percentage = (75π/8 / 567π) * 100
Simplifying the equation:
Percentage = (75/8 / 567) * 100
Percentage = (75/8) * (1/567) * 100
Percentage = (75/567) * 100
Percentage = 13.23 (rounded to the nearest hundredth)
Therefore, approximately 13.23% of the total pizza surface area was covered by the pepperoni.