Problem 63 - Thinking Mathematically

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I paid \(\$ 10\) for a pizza, so I would expect to pay approximately \(\$ 20\) for the same kind of pizza with twice the radius.

Step-by-Step Solution

Answer

The statement does not make sense. At the same cost per unit area, the larger pizza should cost around \$40, not \$20.

Step 1: Determine the Area of the Original Pizza

To compare the expected prices of the pizzas, we first need to calculate the area of the original pizza. Without knowing the actual radius, we can let the radius of the original pizza be \(r\). So, the area of the original pizza, \(A1\) using the formula \(A = \pi r^2\) would be: \(A1 = \pi r^2\)

Step 2: Determine the Area of the Larger Pizza

The larger pizza has twice the radius of the original pizza. That means the new radius is \(2r\). So, the area of the larger pizza, \(A2\) would be: \(A2 = \pi (2r)^2 = 4\pi r^2\). So, the larger pizza has four times the area of the original pizza.

Step 3: Compare the Costs

If the cost of pizza is directly proportional to its area, the larger pizza with four times the area of the original pizza should cost four times as much. So, if the original pizza costs \$10, we would expect the larger pizza to cost approximately \$40, not \$20 like the statement claims.

Key Concepts

Understanding Geometry

Geometry is an essential branch of mathematics that studies the properties and relationships of points, lines, surfaces, and solids. In this particular exercise, we focus on the geometry of a circle, which is fundamental when dealing with pizzas or any circular objects. Knowing basic circle geometry is crucial:

• Radius: The distance from the center of the circle to any point on its circumference.
• Diameter: Twice the radius, extending from one point on the circumference through the center to another point on the circumference.
• Area: The space enclosed within the circle's boundary, calculated using the formula \(A = \pi r^2\).

Understanding these basics helps us make sense of how changing the radius impacts the area.

Basics of Area Calculation

Calculating the area of a circle is a pivotal part of understanding how size increase affects cost. To find the area, we use the formula \(A = \pi r^2\), where \(r\) is the radius. If the radius of a circle is doubled, the new radius becomes \(2r\). In practice, this means: \[ A2 = \pi (2r)^2 = 4\pi r^2 \] This shows that the area doesn't just double, but quadruples. Therefore, for a pizza with double the radius, the area expands significantly, impacting corresponding calculations, such as price, if directly proportional to the area.

Exploring Direct Proportionality

Direct proportionality is a mathematical concept where one quantity increases in direct relation to another. In this scenario, we consider the relationship between the pizza's area and its cost.

• Concept: If the area of a pizza increases, the cost should increase proportionally if all else remains constant.

Using this concept, since the area of the larger pizza is four times that of the original, the cost should also increase four times. That’s why doubling the radius results not in a doubling of the price, but in a price equal to four times the original cost, assuming direct proportionality between area and cost.

Step-by-step Problem Solving

Approaching mathematical problems requires logical steps, just like in the pizza scenario. Identifying relationships and calculations systematically can make solutions clearer. Here are the steps broken down:

• Identify what's being compared: here, area and cost.
• Calculate both areas: using \(A = \pi r^2\) for different scenarios.
• Understand cost implications: use proportionality to determine expected costs.

This methodical approach not only simplifies the problem but ensures no details are overlooked, enhancing the precision of your answer.

Mathematical Explanation Made Simple

Creating a clear and concise mathematical explanation is key in conveying complex ideas easily. In this exercise, we:

• Start with basic geometry to identify essential elements like radius and area.
• Calculate areas with the correct formula to see the exponential increase due to changes in dimensions.
• Apply logical reasoning, such as direct proportionality, to relate the physical size to cost implications clearly.

This fundamental approach helps demystify the mathematics behind real-world problems, making it less daunting and more understandable for learners at all levels.