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When we talk about a pizza's size, what we're really referring to is its **surface area**. To find this area, we use the formula for the area of a circle: \( \pi r^2 \), where \( r \) is the radius. However, the pizza sizes given are measured in diameters, so we must first divide the diameter by 2 to find the radius. For example, a 9-inch pizza has a radius of 4.5 inches (since 9 divided by 2 is 4.5). Substituting this value into the formula, we get: \( \pi (4.5)^2 = 63.62 \) square inches.
In general, understanding the surface area helps us understand how much actual pizza we get. Larger pizzas might seem more expensive at first, but evaluating their surface area gives a clearer picture of their value. By doing this, we can easily compare different pizza sizes to see which provides more pizza.

• 9-inch pizza: 63.62 square inches
• 12-inch pizza: 113.10 square inches
• 15-inch pizza: 176.71 square inches

Cost analysis is crucial when deciding which pizza size gives the best value for money. We determine this by calculating the **cost per square inch**. This involves dividing the total price of the pizza by its surface area. For instance, the 9-inch pizza costs \\(8.78, and with a surface area of 63.62 square inches, the cost per square inch is \( \\)8.78 / 63.62 \approx \\(0.138 \). Likewise, for the larger sizes, we repeat the calculation:

• 12-inch pizza: \( \\)11.78 / 113.10 \approx \\(0.104 \) per square inch
• 15-inch pizza: \( \\)14.18 / 176.71 \approx \$0.080 \) per square inch

By analyzing these costs, we discover the true price of each square inch of pizza. Smaller pizzas may initially look cheaper, but they can be more expensive per square inch. Therefore, through cost analysis, we can evaluate which size is most economical.

Optimization problems occur frequently in everyday decision-making, such as finding the best pizza size for your money. In this context, optimization involves comparing different options to achieve the best outcome; here, it's about minimizing the cost per square inch of pizza.
The 15-inch pizza emerges as the most optimized choice, priced at \\(14.18 but only costing \( \\)0.080 \) per square inch. This makes it the best buy, as it offers the most pizza for your money compared to the smaller sizes which have higher costs per square inch.
Through this exercise, we learn the importance of optimization in practical decisions. Understanding the principles behind calculating cost and value can help make smarter purchasing decisions. So, next time you're buying a pizza, consider the optimization problem to help you get the best bang for your buck!