A 6 inch personal pizza has 620 calories, with 240 of those from fat. A 16 inch pizza is cut into 8 slices. Estimate the number of calories in one slice of a 16 inch pizza. Hint... must use Area = Pi*r^2 proportions to solve this problem.. DO NOT SOLVE BY PIECE.
R/r = 16/6 = 8/3
A/a = (8/3)^2
1/8 * (8/3)^2 * 240 = ____
To estimate the number of calories in one slice of a 16 inch pizza, we can use the proportions of the areas of the 6-inch and 16-inch pizzas.
The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.
Let's assume that the 6-inch pizza has a radius of r1 inches and the 16-inch pizza has a radius of r2 inches.
The area of the 6-inch pizza is A1 = πr1^2.
The area of the 16-inch pizza is A2 = πr2^2.
We can set up a proportion using the ratios of the areas of the two pizzas:
A1/A2 = (πr1^2)/(πr2^2) = (r1^2)/(r2^2)
To find the ratio of the radii, we use the fact that the 16-inch pizza is cut into 8 equal slices, so each slice will have a central angle of 45 degrees (360 degrees / 8 slices = 45 degrees).
The radius of each slice can be found using the formula:
r_slice = r2 * sin(central angle/2)
r_slice = r2 * sin(45/2)
r_slice = r2 * sin(22.5)
Now we can set up the proportion:
A1/A2 = (r1^2)/(r2^2)
Substituting the radii:
(πr1^2)/(πr2^2) = (r1^2)/((r2 * sin(22.5))^2)
To solve for r2, we can rearrange the equation:
r2^2 = r1^2 / (sin(22.5))^2
Taking the square root of both sides, we get:
r2 = sqrt(r1^2 / (sin(22.5))^2)
Now we can substitute the given values:
r1 = 3 inches (radius of the 6-inch pizza)
r2 = sqrt(3^2 / (sin(22.5))^2) = 11.66 inches (approximately)
The radius of a slice of the 16-inch pizza is approximately 11.66 inches.
Now, we can calculate the area of one slice of the 16-inch pizza:
A_slice = π * r_slice^2
A_slice = π * (11.66)^2
A_slice ≈ 426.21 square inches
Finally, we can estimate the number of calories in one slice of the 16-inch pizza by using the ratio of calories per square inch of both pizzas:
Calories per square inch of the 6-inch pizza = 620 calories / (π * (3)^2) ≈ 23.37 calories/square inch
Estimated calories in one slice of the 16-inch pizza = 23.37 calories/square inch * 426.21 square inches ≈ 9947.87 calories
Therefore, the estimated number of calories in one slice of the 16-inch pizza is approximately 9947.87 calories.
To estimate the number of calories in one slice of a 16-inch pizza, we can use proportions and the concept of area to make an approximation. Here's how you can do it:
1. First, calculate the area of the 6-inch pizza using the formula A = πr^2, where A represents the area and r is the radius.
For the 6-inch pizza: A1 = π * (6/2)^2 = 28.27 square inches
2. Next, calculate the area of the 16-inch pizza using the same formula.
For the 16-inch pizza: A2 = π * (16/2)^2 = 201.06 square inches
3. Since the 16-inch pizza is cut into 8 slices, each slice will have an equal portion of the total area. Therefore, we can find the area of one slice by dividing the total area by the number of slices.
Area per slice = A2 / 8 = 201.06 / 8 = 25.13 square inches per slice
4. Now, we can use the proportion of the areas to estimate the number of calories in one slice of the 16-inch pizza.
proportion of areas: (A1 / A2) = (calories per slice of 6-inch pizza / calories per slice of 16-inch pizza)
calories per slice of 16-inch pizza = (A1 / A2) * calories per slice of 6-inch pizza
Let's substitute the given values into the equation:
calories per slice of 16-inch pizza = (28.27 / 25.13) * 620 = 696.38 (approx.)
Therefore, an estimated number of calories in one slice of a 16-inch pizza is approximately 696.38 calories.