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In the figure,
O is the center of the circle. If the area of the shaded region is
42 pi, what is the diameter of the circle?

In the figure, $O$ is the center of the circle. If the area of the shaded region is $42 \pi$ , what is the diameter of the circle?

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Solve proportions: word problems

Full solution

Q. In the figure,

O

is the center of the circle. If the area of the shaded region is

42 \pi

, what is the diameter of the circle?

Circle Area Formula: We know the formula for the area of a circle is $A = \pi \cdot r^2$ , where $A$ is the area and $r$ is the radius.
Setting up Equation: Given the area of the shaded region is $42 \pi$ , we set up the equation $42 \pi = \pi \cdot r^2$ .
Solving for Radius: We can cancel $\pi$ from both sides of the equation, which leaves us with $42 = r^2$ .
Correcting Mistake: To find the radius, we take the square root of both sides, so $r = \sqrt{42}$ .
Correcting Mistake: To find the radius, we take the square root of both sides, so $r = \sqrt{42}$ .But wait, we made a mistake. We should have simplified $42$ to $6 \times 7$ before taking the square root. Let's correct that.

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