A Circular Cookie Cake Costs $15.70. If The Diameter Of The Cookie Cake Is 10 Inches, What Is The Approximate Cost Per Square Inch Of The Cookie Cake? Use Π = 3.14.

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Introduction

In this article, we will delve into a practical mathematical problem involving a circular cookie cake. Understanding how to calculate the cost per square inch of a circular item can be useful in various real-life scenarios, from baking and catering to everyday shopping decisions. This exercise combines basic geometry concepts, such as the area of a circle, with simple arithmetic to arrive at a solution. Our main goal is to determine the approximate cost per square inch of a cookie cake given its total cost and diameter. This involves a step-by-step process that first calculates the area of the cookie cake and then divides the total cost by this area. The problem not only reinforces mathematical skills but also provides a tangible example of how math is applied in everyday contexts.

Understanding the Problem

Let's begin by restating the problem to ensure we fully grasp what is being asked. We have a circular cookie cake that costs $15.70. The diameter of the cookie cake is given as 10 inches. Our task is to find the approximate cost per square inch of this cookie cake. This means we need to determine how much each square inch of the cake costs. To solve this, we'll need to use the formula for the area of a circle, which is πr², where r is the radius of the circle, and π (pi) is approximately 3.14. The diameter is the distance across the circle through the center, so the radius is half of the diameter. Once we calculate the area, we can divide the total cost by the area to find the cost per square inch. This problem highlights the connection between geometry and practical calculations, demonstrating how mathematical concepts are applied in everyday situations. The challenge lies in accurately applying the formula for the area of a circle and then performing the division to find the cost per unit area. By breaking down the problem into these steps, we can systematically arrive at the solution.

Step 1 Calculate the Radius

The first step in solving this problem is to determine the radius of the circular cookie cake. The radius is the distance from the center of the circle to any point on its edge. We are given the diameter, which is the distance across the circle through the center. The relationship between the radius (r) and the diameter (d) is simple: the radius is half of the diameter. Mathematically, this is expressed as r = d/2. In our case, the diameter of the cookie cake is 10 inches. To find the radius, we divide the diameter by 2: r = 10 inches / 2 = 5 inches. Therefore, the radius of the cookie cake is 5 inches. This simple calculation is crucial because the radius is a key component in the formula for the area of a circle, which we will use in the next step. Understanding this relationship between diameter and radius is fundamental in geometry and essential for solving problems involving circles. This step sets the foundation for the subsequent calculations and ensures we have the correct value to proceed. With the radius determined, we can now move on to calculating the area of the cookie cake.

Step 2 Calculate the Area

Now that we have the radius, we can calculate the area of the circular cookie cake. The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is approximately 3.14, and r is the radius. We have already determined that the radius of the cookie cake is 5 inches. Plugging this value into the formula, we get A = 3.14 * (5 inches)². To calculate (5 inches)², we multiply 5 inches by itself, which equals 25 square inches. So, the equation becomes A = 3.14 * 25 square inches. Multiplying 3.14 by 25 gives us 78.5 square inches. Therefore, the area of the cookie cake is approximately 78.5 square inches. This calculation is crucial because the area represents the total surface of the cookie cake, which we need to determine the cost per square inch. The formula for the area of a circle is a fundamental concept in geometry, and its accurate application is essential for solving this problem. With the area calculated, we are now one step closer to finding the cost per square inch. The next step involves using the total cost of the cookie cake and the calculated area to find the cost per unit area.

Step 3 Calculate the Cost Per Square Inch

With the area of the cookie cake calculated, we can now determine the cost per square inch. To find the cost per square inch, we divide the total cost of the cookie cake by its area. We know that the cookie cake costs $15.70 and the area is approximately 78.5 square inches. Therefore, the cost per square inch is calculated as: Cost per square inch = Total cost / Area. Plugging in the values, we get: Cost per square inch = $15.70 / 78.5 square inches. Performing this division, we find that the cost per square inch is approximately $0.20. This means that each square inch of the cookie cake costs about 20 cents. This calculation provides a practical understanding of the value of the cookie cake in terms of its surface area. By dividing the total cost by the area, we are essentially distributing the cost evenly across each square inch of the cake. This is a useful concept in various scenarios, such as pricing baked goods or comparing the value of different-sized items. The final answer, $0.20 per square inch, gives us a clear understanding of the cookie cake's cost in relation to its size. This concludes the solution to our problem, providing a tangible example of how mathematical calculations can be applied in real-world situations.

Conclusion

In this article, we successfully calculated the approximate cost per square inch of a circular cookie cake. We began by understanding the problem, identifying the given information (total cost and diameter) and the desired outcome (cost per square inch). We then systematically worked through the problem, starting with calculating the radius from the given diameter. Next, we used the radius to calculate the area of the cookie cake using the formula A = πr². Finally, we divided the total cost of the cookie cake by its area to find the cost per square inch. The result, approximately $0.20 per square inch, provides a clear understanding of the value of the cookie cake in relation to its size. This exercise demonstrates the practical application of mathematical concepts, such as the area of a circle, in everyday scenarios. By breaking down the problem into smaller, manageable steps, we were able to arrive at the solution efficiently and accurately. This approach highlights the importance of a structured problem-solving strategy in mathematics and its relevance in real-world contexts.