Final answer:
The maximum speed at which you can make a turn of radius 10 m without slipping on a slippery surface is 4.0 m/s, given that you are on the edge of slipping at this speed. Thus, the correct answer is 4.0 m/s.
Step-by-step explanation:
Understanding Centripetal Force and Friction
When running on a slippery surface and attempting to make a turn without slipping, the centripetal force required for the turn is provided by the frictional force between your feet and the ground. This frictional force is the product of the coefficient of friction and the normal force (which is equal to your weight if we ignore any incline). Since you are on the verge of slipping at a speed of 4.0 m/s for a 10 m radius turn, this is the maximum speed at which the existing frictional force can provide the required centripetal force.
Therefore, the maximum speed at which you can make the turn without slipping is 4.0 m/s. Any speed lower than this can also be maintained without slipping, given that it would result in a lesser demand for centripetal force. Hence, options 3.0 m/s, 4.0 m/s, and 5.0 m/s are technically correct, with 4.0 m/s being the highest possible speed.
However, if the question implies finding the single maximum speed, the correct answer is option 2) 4.0 m/s. This is because it is the given condition under which you are just able to make the turn without slipping. Increasing the speed beyond this would require more friction than is available, leading to slipping.