Water flows completely through a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds: $$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$ (Here, $k$ is a positive constant, the unit of length is m, and the unit of speed is m/s.) When the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center. What is the value of $a$? [3 points] (1) $\frac { 39 } { 23 }$ (2) $\frac { 37 } { 23 }$ (3) $\frac { 35 } { 23 }$ (4) $\frac { 33 } { 23 }$ (5) $\frac { 31 } { 23 }$
Water flows completely through a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds:
$$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$
(Here, $k$ is a positive constant, the unit of length is m, and the unit of speed is m/s.)
When the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center. What is the value of $a$? [3 points]\\
(1) $\frac { 39 } { 23 }$\\
(2) $\frac { 37 } { 23 }$\\
(3) $\frac { 35 } { 23 }$\\
(4) $\frac { 33 } { 23 }$\\
(5) $\frac { 31 } { 23 }$