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Chapter 20: Chapter 20

Problem 381

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The area of a circular region is known to be \(154 \mathrm{sq}\). in. a) What is the diameter of the circle? b) What is its circumference? (Assume \(\pi=22 / 7 .\) )

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a) The diameter of the circle is 14 inches. b) The circumference of the circle is 44 inches.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Use the area formula to find the radius

The formula for the area of a circle is \(A = πr^2\), where A is the area and r is the radius. We are given that the area A is 154 square inches, and we are asked to assume that \(\pi = \frac{22}{7}\). To find the radius, we will rearrange the area formula to solve for r: \(r = \sqrt{\frac{A}{π}}\) Now, plug in the values for A and \(\pi\) and solve for the radius: \(r = \sqrt{\frac{154}{\frac{22}{7}}}\)

Step 2: Simplify the expression

Next, we will simplify the expression inside the square root: \(r = \sqrt{\frac{154\cdot 7}{22}}\) \(r = \sqrt{\frac{1078}{22}}\) To continue simplifying, we'll divide 1078 by 22: \(r = \sqrt{49}\)

Step 3: Calculate the radius

Now that we have simplified our expression, we can find the radius by taking the square root of 49: \(r = 7\) So, the radius of the circle is 7 inches.

Step 4: Find the diameter

The diameter (d) of a circle is twice its radius: \(d = 2r\) Now, plug in the value of the radius (7 inches) we found in the previous steps: \(d = 2\cdot 7\) \(d = 14\) So, the diameter of the circle is 14 inches.

Step 5: Find the circumference

The formula for the circumference (C) of a circle is \(C = 2πr\), where r is the radius. We already know the radius is 7 inches, and we are asked to assume that \(\pi = \frac{22}{7}\). Now, plug in the values for r and \(\pi\) and solve for the circumference: \(C = 2\cdot \frac{22}{7} \cdot 7\) \(C = 44\) So, the circumference of the circle is 44 inches. To summarize: a) The diameter of the circle is 14 inches. b) The circumference of the circle is 44 inches.

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