Final answer:
The angle that the chalk line makes at time 1.88 s is 7.8054 radians, which doesn't match any of the given options, suggesting an error in the question or answer choices.
Step-by-step explanation:
The angle made by the chalk line on the wheel at time t can be found by using the kinematic equation for rotational motion: \(\theta = \omega_{0}t + \frac{1}{2}\alpha t^2\), where \(\theta\) is the angle in radians, \(\omega_{0}\) is the initial angular speed, \(\alpha\) is the angular acceleration, and t is the time.
Given an initial angular speed of 2.57 rad/s, an angular acceleration of 1.69 rad/s², and a time of 1.88 s, we can calculate:
\(\theta = (2.57 \text{ rad/s})(1.88 \text{ s}) + \frac{1}{2}(1.69 \text{ rad/s}^2)(1.88 \text{ s})^2\)
\(\theta = 4.8316 + 2.9738\)
\(\theta = 7.8054 \text{ rad}\)
The closest answer to 7.8054 rad provided in the options is Option A) 4.38 radians, which indicates a possible error in the question or the answer choices. If this were a test or homework, the student should verify the calculations and options with the instructor.