Question

Diego runs back and forth along a straight track. During the time interval 0≤t≤30 seconds, Diego’s velocity, in feet per second, is modeled by the function v given by v(t)=200sin(t275)t+1.

(a) What is the first time, t1, that Diego changes direction? Find Diego’s average velocity over the time interval 0≤t≤t1.
Question 2
(b) At time t=30, how far is Diego from his position at time t=0 ?
Question 3
(c) Find the total distance that Diego runs during the 30 seconds from time t=0 to time t=30.
Question 4
(d) Let a(t) represent Diego’s acceleration during the time interval 0

Answer 1

To determine when Diego changes direction, the average velocity, and the distance covered, one must analyze the velocity function v(t). Changes in direction are indicated by the function crossing zero, and average velocity is determined by integrating v(t). Distance is calculated by integrating the absolute value of v(t).

Step-by-step explanation:

The question posted is missing a critical part of the function that describes Diego's velocity, but based on the given information from the challenge problems, similar steps can be followed to determine when Diego changes direction, the average velocity over a given interval, and the distance covered in a specified time frame. In general, a change in direction occurs when the velocity function v(t) crosses from positive to negative or vice versa, which corresponds to the times when v(t) = 0. Average velocity over an interval [a, b] can be found by integrating the velocity function over that interval and dividing by the time difference (b - a).

For the distance covered by Diego from time t=0 to t=30, we would integrate the absolute value of the velocity function over the interval [0, 30], since distance is a scalar quantity and does not depend on direction. The acceleration function a(t) is the derivative of the velocity function v(t), and it provides information about Diego's acceleration at any given point in time. To find these values, one would differentiate the given velocity function and evaluate at the specified times.