Question

Write a program that calculates the cost per square inch of a circular pizza, given its diameter and price. The formula for area is \(A=\pi r^{2}\).

Short answer

Divide the pizza's price by its area (using radius from diameter).

Step-by-step solution

Understand the Problem

We need to calculate the cost per square inch of a pizza. The inputs are the diameter of the pizza and its price. We'll use the formula for the area of a circle, which requires the radius.

Calculate the Radius

The radius is half of the diameter. If the diameter is `d`, then the radius `r` is calculated as: \( r = \frac{d}{2} \).

Calculate the Area of the Pizza

Using the formula for the area of a circle, \( A = \pi r^2 \), we calculate the area. Since \( r \) is the radius from step 2, substitute \( r \) into the formula to find the area.

Compute the Cost Per Square Inch

To find the cost per square inch, divide the price of the pizza by its area: \( \text{Cost per square inch} = \frac{\text{Price}}{A} \).

Program the Solution

Write the program code. Below is a sample in Python: ```python import math def cost_per_square_inch(diameter, price): radius = diameter / 2 area = math.pi * (radius ** 2) cost_per_inch = price / area return cost_per_inch ``` This function will take diameter and price as input and return the cost per square inch.

Key concepts

Circle Area Calculation

Calculating the area of a circle is crucial in various mathematical and real-life applications, like determining the cost efficiency of a pizza per square inch. The formula to find the area of a circle is given by \( A = \pi r^2 \), where \( A \) represents the area, and \( r \) is the radius of the circle.

• The symbol \( \pi \) (Pi) is a constant approximately equal to 3.14159.
• The radius \( r \) is simply half of the circle's diameter.
• The exponent \( 2 \) indicates that the radius is squared.

To calculate the area, you should first find the radius, then plug this number into the formula. For example, if a pizza's diameter is 12 inches, the radius would be 6 inches (half of 12).
Now, let's compute the area:\[A = \pi \times 6^2 = \pi \times 36 \approx 113.1 \, \text{square inches}\]This step is essential for understanding how much surface area you're getting when you purchase a circular object like a pizza.

Radius and Diameter

The relationship between the radius and diameter of a circle is straightforward yet fundamental to solving problems involving circles. The diameter is the longest distance across the circle, passing through its center, while the radius is half this length.

• The formula to find the radius from the diameter is: \( r = \frac{d}{2} \).
• Conversely, you can find the diameter from the radius by doubling it: \( d = 2r \).
• Understanding these relationships helps simplify calculations for area or circumference.

Knowing whether you're given the radius or diameter initially is crucial, as it impacts how you proceed with further calculations, such as finding the area. In programming, clear identification of these values is important to ensure accuracy in your computations.

Cost Efficiency in Programming

Cost efficiency in programming is about finding the most efficient way to solve a problem while balancing resources, such as time and processing power. When calculating the cost efficiency of a pizza, the goal is to determine how much value for money you're getting per square inch.

• Start by understanding the inputs: the pizza's diameter and price must be known.
• Calculate the pizza's area to figure out how much pizza you're getting.
• Compute the cost by using the formula: \( \text{Cost per square inch} = \frac{\text{Price}}{A} \).

In Python, these steps can be programmed efficiently to quickly assess the value proposition of different pizza sizes and prices. Writing clear and efficient code helps minimize errors and maximizes clarity, ensuring that anyone using the program can easily modify or extend it in the future.