$\text { Find the arc length from } ( 0,4 ) \text { clockwise to } ( 2,2 \sqrt { 3 } ) \text { along the circle } x ^ { 2 } + y ^ { 2 } = 16 \text {. }$ Round your answer to four decimal places.
A)4.1888
B)0.2618
C)0.1309
D)1.8546
E)2.0944

Question

Quizplus · Accepted Answer

The circle $x^2 + y^2 = 16$ has a radius of 4. The points $(0,4)$ and $(2,2\sqrt{3})$ form an angle with the center of the circle. To find the arc length, we first find the angle in radians between these two points. The angle can be found using the dot product or by recognizing the special triangle formed. The point $(2,2\sqrt{3})$ corresponds to a 30-degree angle (or $\pi/6$ radians) in standard position, and $(0,4)$ is at a 90-degree angle (or $\pi/2$ radians). The difference in angles is $\pi/2 - \pi/6 = \pi/3$ radians. The arc length is then the radius times the angle in radians: $4 \times \pi/3 = 4\pi/3$. When calculated, $4\pi/3$ is approximately $4.18879$, which rounds to $4.1888$, but since the movement is clockwise from $(0,4)$ to $(2,2\sqrt{3})$, we actually cover three-fourths of the circle's circumference minus the arc length calculated, leading to the correct calculation for the clockwise direction. The circumference of the circle is $2\pi r = 2\pi(4) = 8\pi$, and three-fourths of that is $6\pi$. Subtracting the previously calculated arc ($4\pi/3$) from $6\pi$ gives us $6\pi - 4\pi/3 = 18\pi/3 - 4\pi/3 = 14\pi/3$, which was not directly calculated in the initial explanation, leading to a misunderstanding in the final step. The correct approach to find the clockwise arc length directly involves calculating the angle and using the formula for arc length ($s = r\theta$), where $\theta$ should be correctly identified for the clockwise direction. Given the misunderstanding in the calculation process, the correct answer should reflect the arc length for the segment specified, which is $2\pi/3$ of the circle's circumference, not the entire calculation provided. The correct answer, based on the arc length directly from $(0,4)$ to $(2,2\sqrt{3})$, should be calculated as $s = r\theta = 4 \times (\pi/3)$, which gives $4\pi/3$, but the correct interpretation of the question leads to identifying the arc length for the specified segment without the need for subtracting from the total circumference. The error in explanation led to confusion about the correct process for determining the clockwise arc length. The correct answer should reflect the direct calculation of the arc length for the specified path, which involves correctly identifying the angle in radians between the two points and using the arc length formula without unnecessary subtraction from the total circumference.

Round Your Answer to Four Decimal Places