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Problem 381

# 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.
See the step by step solution

## 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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