Problem 37 The radius of circle \(B\) is 2.... [FREE SOLUTION]

Chapter 10: Problem 37

The radius of circle \(B\) is 2.5 times the radius of circle \(A .\) If the area of circle \(A\) is 8 square yards, what is the area of circle \(B ?\) (CIRCLE CANNOT COPY)

Short Answer

Expert verified

The area of circle B is 50 square yards.

Step by step solution

Understanding the Relationship

First, we identify that the radius of circle B is 2.5 times that of circle A. Thus, if the radius of circle A is denoted as \( r_A \), then the radius of circle B, \( r_B \), can be written as \( r_B = 2.5 \times r_A \).

Finding the Radius of Circle A

We know the area of circle A is given by \[\text{Area of circle A} = \pi \times (r_A)^2 = 8\]Solving for \( r_A \), we have:\[(r_A)^2 = \frac{8}{\pi}\]Thus, the radius of circle A is \[ r_A = \sqrt{\frac{8}{\pi}} \].

Calculating the Radius of Circle B

Substituting the expression for \( r_A \) into the relationship for \( r_B \), we find:\[ r_B = 2.5 \times \sqrt{\frac{8}{\pi}}\]

Finding the Area of Circle B

The area of circle B is given by the formula\[\text{Area of circle B} = \pi \times (r_B)^2\]Substituting the value of \( r_B \) we found:\[\text{Area of circle B} = \pi \times \left(2.5 \times \sqrt{\frac{8}{\pi}}\right)^2\]Simplifying further, we have:\[\text{Area of circle B} = \pi \times \left(6.25 \times \frac{8}{\pi}\right) = 6.25 \times 8 = 50\]Therefore, the area of circle B is 50 square yards.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius Calculation

The radius of a circle is a crucial dimension that can unlock other geometric properties of the circle, such as its area and circumference. In our problem, we deal with two circles, A and B. To calculate the radius of each circle, we started with circle A. For any circle, the area can be given by the formula:

Area = \(\pi \times r^2\)

Given, the area of circle A is 8 square yards. We can rearrange the formula to solve for the radius of circle A, \(r_A\):

\(\pi \times (r_A)^2 = 8\)
\((r_A)^2 = \frac{8}{\pi}\)
\(r_A = \sqrt{\frac{8}{\pi}}\)

Now for circle B, the radius is specified to be 2.5 times that of circle A, or \(r_B = 2.5 \times r_A\).Knowing how to calculate the radius is vital as it leads to determining other aspects of circle geometry.

Circle Geometry

Circle geometry is the study of properties and measures of a circle, focusing primarily on parameters such as radius, diameter, circumference, and area. One of the foundational aspects of geometry is understanding how these elements are related. Let鈥檚 break these down:

Radius (r): The distance from the center of the circle to any point on its circumference.
Diameter (d): Twice the radius, or the longest distance across the circle, passing through the center.
Area (A): Calculated as \( \pi r^2 \), which measures the space contained within the circle.
Circumference (C): The total distance around the circle, calculated as \( 2\pi r \).

With these formulae in mind, if we know the radius, we can easily find other properties, which makes understanding radius absolutely central to solving geometric problems involving circles.

Problem-Solving Steps

Solving problems in mathematics often involves a systematic approach to bridge known concepts to the unknown quantities you wish to determine. In this area calculation problem, we followed specific steps:

Step 1: Understanding the Relationship: Recognize the relation between the radius of circle B and circle A. This fundamental observation laid the groundwork for further calculations.
Step 2: Calculating Radius of Circle A: Use the circle area formula and given area to derive the radius of circle A.
Step 3: Calculating the Radius of Circle B: Utilize the relationship that \( r_B = 2.5 \times r_A \) to find circle B's radius.
Step 4: Calculating the Area of Circle B: Substitute the expression for \( r_B \) back into the area formula to find circle B's area.

By walking through each step methodically, you can systematically tackle and solve complex geometric problems with confidence, ensuring you understand each part of the puzzle.

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91影视

The radius of circle \(B\) is 2.5 times the radius of circle \(A .\) If the area of circle \(A\) is 8 square yards, what is the area of circle \(B ?\) (CIRCLE CANNOT COPY)

Short Answer

Step by step solution

Understanding the Relationship

Finding the Radius of Circle A

Calculating the Radius of Circle B

Finding the Area of Circle B

Key Concepts

Radius Calculation

Circle Geometry

Problem-Solving Steps

One App. One Place for Learning.

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