Step 1: Set up the two cases.
The same body slides down the same slope twice. First the slope is smooth, then it is rough. The slope length is the same both times, but the rough one is slower because friction fights the motion.
Step 2: Acceleration on each slope.
On the smooth slope only gravity acts along the incline. \[ a_1 = g\sin\theta \] On the rough slope friction is subtracted. \[ a_2 = g(\sin\theta - \mu\cos\theta) \]
Step 3: Use the same distance idea.
Starting from rest, distance is $s = \tfrac12 a t^2$. Since the distance is equal in both cases, and times are $T$ and $3T$, \[ a_1 T^2 = a_2 (3T)^2 \] This gives $a_1 = 9a_2$.
Step 4: Plug in the accelerations.
\[ g\sin\theta = 9\,g(\sin\theta - \mu\cos\theta) \] Cancel $g$ from both sides.
Step 5: Solve for the friction coefficient.
\[ \sin\theta = 9\sin\theta - 9\mu\cos\theta \] \[ 9\mu\cos\theta = 8\sin\theta \] \[ \mu = \frac89 \tan\theta \]
Step 6: Use the given angle.
Here $\tan\theta = \tfrac{9}{16}$. \[ \mu = \frac89 \times \frac{9}{16} = \frac12 = 0.5 \] \[ \boxed{0.5} \]