the sector of a circle shown at left has center at $o$ . The radius of the circular sector is $10$ and the area of the sector is $75$ . What is the length of the arc $xyz$ ?

Full solution

Q. the sector of a circle shown at left has center at

o

. The radius of the circular sector is

10

and the area of the sector is

75

. What is the length of the arc

xyz

Sector Area Formula: The area of a sector of a circle is given by the formula $A = \frac{1}{2} r^2 \theta$ , where $A$ is the area of the sector, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. $\newline$ Given: $\newline$ Radius ( $r$ ) = $10$ $\newline$ Area of the sector ( $A$ ) = $75$ $\newline$ We need to find the central angle $\theta$ in radians first. $\newline$ Using the formula for the area of a sector, we can solve for $\theta$ : $\newline$ $75 = \frac{1}{2} \times 10^2 \times \theta$
Calculate Central Angle: Now, we calculate the value of $\theta$ : $\newline$ $75 = \frac{1}{2} \times 100 \times \theta$ $\newline$ $75 = 50 \times \theta$ $\newline$ $\theta = \frac{75}{50}$ $\newline$ $\theta = \frac{3}{2}$ radians
Arc Length Formula: The length of the arc ( $L$ ) of a sector is given by the formula $L = r \theta$ , where $L$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians. $\newline$ We have: $\newline$ Radius ( $r$ ) = $10$ $\newline$ Central angle ( $\theta$ ) = $\frac{3}{2}$ radians $\newline$ Now we can calculate the length of the arc $L$ : $\newline$ $L = 10 \times \frac{3}{2}$
Calculate Arc Length: Calculating the length of the arc $L$ : $\newline$ $L = 10 \times \frac{3}{2}$ $\newline$ $L = 15$ $\newline$ The length of the arc XYZ is $15$ units.