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Chapter 9: Problem 6

Find the area of a circle whose radius is 7 inches. Use \(\pi=3.14\) A. 21.98 in. \(^{2}\) B. 43.96 in \(^{2}\) C. 153.86 in \(^{2}\) D. 615.44 in \(^{2}\)

Short Answer

Expert verified
The area of the circle is 153.86 in \(^2\), so option C is correct.

Step by step solution

01

Understand the Formula

To find the area of a circle, we use the formula: \( A = \pi r^2 \), where \( A \) is the area, \( \pi \) is a constant approximately equal to 3.14, and \( r \) is the radius of the circle.
02

Substitute the Given Radius

Given the radius of the circle is 7 inches, substitute \( r = 7 \) into the formula: \( A = 3.14 \times 7^2 \).
03

Calculate the Square of the Radius

Square the radius: \( 7^2 = 49 \). So, the formula becomes: \( A = 3.14 \times 49 \).
04

Multiply by Pi

Multiply 49 by \( \pi \) (3.14): \( A = 3.14 \times 49 = 153.86 \).
05

Conclusion and Selection of the Correct Option

The calculated area of the circle is \( 153.86 \text{ in}^2 \). Compare with the options given: the correct option is C. 153.86 \( \text{ in}^2 \).

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Key Concepts

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

Understanding the Radius
The radius is a fundamental concept when discussing circles. It is the distance from the center of the circle to any point on its edge.
The radius plays a crucial role in calculating both the circumference and the area of a circle.
  • If you know the diameter of a circle, you can find the radius by dividing the diameter by two.
  • The radius is usually represented by the letter \( r \) in mathematical formulas.
For example, if you're given a circle with a radius of 7 inches, you're working with a circle whose boundary runs 7 inches away from the center. This simple measurement opens the door to calculating the area using a specific formula that relates to another important mathematical constant.
Approximating Pi (\( \pi \))
Pi, often represented by the Greek letter \( \pi \), is a mathematical constant that is critical in calculations involving circles. While pi is an irrational number approximately equal to 3.14159, for most practical purposes, it is often approximated as 3.14.
  • Using 3.14 as an approximation makes calculations more straightforward without significantly impacting accuracy for common tasks.
  • Pi represents the ratio of a circle's circumference to its diameter, and it is the same for all circles.
Whether you're calculating the circumference, area, or another property of the circle, pi is essential. In the exercise given, \( \pi = 3.14 \) is used to simplify the arithmetic, while still ensuring the result is reasonably close to the actual value.
Using the Mathematical Formula for Area
To find the area of a circle, the mathematical formula \( A = \pi r^2 \) is used. This formula indicates that the area \( A \) is equal to pi multiplied by the square of the radius \( r \).
  • This formula derives from the fundamental geometry of circles and is essential for solving any problems involving a circle's area.
  • Square the radius, then multiply by the approximation of pi to find the area accurately.
In action, for a radius of 7 inches, it might look like this: calculate \( 7^2 = 49 \), then multiply by 3.14 to obtain \( A = 153.86 \).
That's how you derive the area step by step with confidence and understanding.
Building Problem-Solving Skills
Approaching mathematical problems systematically can significantly enhance problem-solving skills. When determining the area of a circle, a structured approach like the step-by-step process presented can be incredibly useful.
  • Start by understanding the problem: recognize what's being asked and identify the components (like the radius).
  • Use the right formula and constants, such as the approximation for pi.
  • Follow each step carefully, ensuring each calculation is checked as you progress.
By practicing this methodical approach, students can develop their critical thinking and analytical skills. Break the problem into smaller, more manageable parts, like substituting values into the formula and performing arithmetic operations, to ensure a clearer path to the correct solution.
As you practice, applying this strategy to various problems will become more intuitive, building broader mathematical competence.

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Most popular questions from this chapter

Find the volume of a cone whose base has a diameter of 12 feet and whose height is 9 feet. Use \(\pi=3.14\) A. \(339.12 \mathrm{ft}^{3}\) B. \(405.06 \mathrm{ft}^{3}\) C. \(36 \mathrm{ft}^{3}\) D. \(1,356.48 \mathrm{ft}^{3}\)

Find the area of a trapezoid whose height is 15 inches and whose bases are 6 inches and 16 inches. A. 93 in. \(^{2}\) B. 165 in. \(^{2}\) C. 330 in. \(^{2}\) D. 168 in \(^{2}\)

Find the length of the hypotenuse of a right triangle whose sides are 180 inches and 112 inches. A. 142 in. B. 292 in. C. 472 in. D. 212 in

Find the area of a triangle whose base is 56 yards and whose height is 45 yards. A. \(1,260\) yd \(^{2}\) B. \(2,520\) yd \(^{2}\) C. 202 yd \(^{2}\) D. 303 yd \(^{2}\)

\- Find the volume of a cube whose side is 6.1 inches in length. A. 18.3 in \(^{3}\) B. 37.21 in. \(^{3}\) C. 24.4 in \(^{3}\) D. 226.981 in. \(^{3}\)
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