B

Step 1: Understand the relationship of angular speeds
For two wheels connected by a belt that does not slip, the relationship between their angular speeds is given by:
$$ rac{r_A}{r_B} = rac{ heta_A}{ heta_B} $$
where $r_A$ and $r_B$ are the radii of wheels A and B, respectively, and $\theta_A$ and $\theta_B$ are their angular displacements.

Step 2: Determine angular acceleration
Given that wheel A has an angular acceleration of $\frac{\pi}{2} \text{ rad/s}^2$, the angular velocity of wheel A at any time $t$ can be given by:
$$ \omega_A = \alpha_A \cdot t = \frac{\pi}{2} t $$

Step 3: Relate angular velocity of wheels A and B
The angular velocity of wheel B ($\omega_B$) can be determined using the radius ratio:
$$ \omega_B = \frac{r_A}{r_B} \cdot \omega_A = \frac{10}{25} \cdot \omega_A = \frac{2}{5} \cdot \omega_A $$
Substituting for $\omega_A$:
$$ \omega_B = \frac{2}{5} \cdot \frac{\pi}{2} t = \frac{\pi}{5} t $$

Step 4: Convert the desired speed of wheel B
To find the time at which wheel B reaches 100 RPM, we need to convert it to radians per second:
$$ 100 \text{ RPM} = \frac{100 \times 2\pi}{60} \text{ rad/s} = \frac{100\pi}{30} \text{ rad/s} = \frac{10\pi}{3} \text{ rad/s} $$

Step 5: Set angular velocity of wheel B equal to desired speed
We set the angular velocity of wheel B equal to the converted speed:
$$ \frac{\pi}{5} t = \frac{10\pi}{3} $$

Step 6: Solve for time $t$
Dividing both sides by $\pi$ gives:
$$ \frac{1}{5} t = \frac{10}{3} $$
Multiplying both sides by 5 gives:
$$ t = \frac{10 \times 5}{3} = \frac{50}{3} \text{ seconds} $$

Final Answer: This corresponds to approximately 16.67 seconds. Thus, wheel B attains a speed of 100 RPM in approximately 16.67 seconds.

Therefore, B.