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In the figure,
O is the center of the circle. The minor arc
AB^(⏜) has a length of
4pi and is
(1)/(8) the circumference of the circle. If the area of the shaded region is
a pi, what is the value of
a ?

In the figure, $O$ is the center of the circle. The minor arc $\widehat{A B}$ has a length of $4 \pi$ and is $\frac{1}{8}$ the circumference of the circle. If the area of the shaded region is $a \pi$ , what is the value of $a$ ?

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Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. In the figure,

O

is the center of the circle. The minor arc

\widehat{A B}

has a length of

4 \pi

and is

\frac{1}{8}

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

a \pi

, what is the value of

a

?

Calculate radius of circle: step_2: Calculate the radius of the circle. $\newline$ The circumference of a circle is given by the formula $C = 2 \cdot \pi \cdot r$ , where $r$ is the radius. $\newline$ $32\pi = 2 \cdot \pi \cdot r$ $\newline$ $r = \frac{32\pi}{2 \cdot \pi}$ $\newline$ $r = 16$
Calculate area of circle: step_3: Calculate the area of the circle. $\newline$ The area of a circle is given by the formula $A = \pi \cdot r^2$ . $\newline$ $A = \pi \cdot 16^2$ $\newline$ $A = \pi \cdot 256$
Calculate area of shaded region: step_4: Calculate the area of the shaded region. $\newline$ The shaded region is the entire circle minus the area of the sector corresponding to the minor arc AB. $\newline$ The sector's area is $(1/8)$ of the circle's area since the arc is $(1/8)$ of the circumference. $\newline$ Sector area = $(1/8) \times \pi \times 256$ $\newline$ Sector area = $32\pi$
Subtract sector area: step_5: Subtract the sector's area from the circle's area to find the area of the shaded region. $\newline$ Shaded area = Circle area - Sector area $\newline$ Shaded area = $\pi \times 256 - 32\pi$ $\newline$ Shaded area = $224\pi$

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