Question

At Wesley's Pizzeria, the small pizza is 4 inches smaller in diameter than the large pizza, whose diameter is 16 inches. If each pizza is cut into eight equal slices, one large slice is approximately what percent larger than one small slice? A. $44\mathrm{\%}$ B. $62\mathrm{\%}$ C. $78\mathrm{\%}$ D. $84\mathrm{\%}$

Short answer

One large slice is approximately 78% larger than one small slice.

Step-by-step solution

Calculate the diameter of the small pizza

The diameter of the large pizza is given as 16 inches. Since the small pizza is 4 inches smaller in diameter, subtract 4 from 16. Therefore, the diameter of the small pizza is 16 - 4 = 12 inches.

Determine the radius of each pizza

The radius is half of the diameter. The radius of the large pizza is 16 / 2 = 8 inches, and the radius of the small pizza is 12 / 2 = 6 inches.

Calculate the area of each pizza

The area of a circle is given by $\pi r^2$. The area of the large pizza is $\pi (8{)}^2=64\pi$ square inches. The area of the small pizza is $\pi (6{)}^2=36\pi$ square inches.

Calculate the area of one slice from each pizza

Each pizza is cut into eight equal slices. So, the area of one slice of the large pizza is $\frac{64\pi }{8}=8\pi$ square inches, and the area of one slice of the small pizza is $\frac{36\pi }{8}=4.5\pi$ square inches.

Calculate the difference in the area between one large slice and one small slice

Subtract the area of one small slice from the area of one large slice: $8\pi -4.5\pi =3.5\pi$ square inches.

Determine the percent increase

To find the percent increase, divide the difference by the area of the small slice and multiply by 100. The percent increase is $\frac{3.5\pi }{4.5\pi }\times 100=\frac{3.5}{4.5}\times 100\approx 77.78\mathrm{\%}$. Rounded, this is approximately 78%.

Key concepts

geometry

Geometry often deals with the properties and measurements of shapes. In the pizzeria problem, we work with circles, understanding their diameters, radii, and areas.
First, remember that the diameter is the length of a line that passes through the center of a circle, touching both sides. The radius is half of the diameter, extending from the center to any point on the circle.
By cutting the pizza (a circle in form) into equal slices, we're essentially working with sectors of the circle. Understanding these fundamental concepts can help us solve the problem.

percent increase

A percent increase compares the difference between a new and an original value to the original value, then expresses this ratio as a percentage. Our task is to determine how much larger a large pizza slice is compared to a small pizza slice.
Here's how you find the percent increase:
1. Subtract the original value (small slice area) from the new value (large slice area) to get the difference.
2. Divide this difference by the original value.
3. Multiply the result by 100 to convert it to a percentage.
In the pizzeria problem, we found the areas of the slices and used these steps to determine the increase in size.

circle area calculation

Calculating the area of a circle is essential in this problem. The formula for the area of a circle is given by: $$\text{Area}=\pi r^2$$.

By substituting the radius values for both the small and large pizzas, we found their respective areas.

For the large pizza, \pi(8^{2}) = 64 \pi square inches.
For the small pizza, \pi(6^{2}) = 36 \pi square inches.
These areas then allowed us to determine the size of each slice and compare them accurately.

mathematical reasoning

Mathematical reasoning involves logically applying mathematical concepts to solve a problem. We use it to translate a real-world scenario, like comparing pizza slice sizes, into a mathematical question.
First, we needed to identify the necessary information: diameters, radii, and how the pizzas are sliced.
Next, apply mathematical steps for calculation, ensuring each step follows logically from the previous one. Determining areas and performing fractional division are all part of applying rigorous, step-by-step reasoning.
Finally, translate the resulting values into understandable measurements, like percentages, to answer the given question.

Through clear, logical steps and calculations, we solve the problem thoroughly and accurately.