At time t in seconds, a particle's distance s(t), in centimeters, from a point is given by s(t) = 4 + 3 sin t. What is the average velocity of the particle from t = pi/3 to t = 7pi/3?
avg velocity=changeDisplacement/time
avg velocitiy=(4+3sin7PI/3-4-3sinPI/3)/(2pi)
figure that out.
To find the average velocity of the particle from time t = π/3 to t = 7π/3, we need to calculate the change in distance divided by the change in time.
Let's start by finding the distance of the particle at t = π/3 and t = 7π/3. We'll plug these values into the equation for s(t) = 4 + 3sin(t).
s(π/3) = 4 + 3sin(π/3)
= 4 + 3(√3/2)
= 4 + (3√3)/2
= (8 + 3√3)/2
= (8√3 + 3)/2
s(7π/3) = 4 + 3sin(7π/3)
= 4 + 3(-√3/2)
= 4 - (3√3)/2
= (8 - 3√3)/2
= (8√3 - 3)/2
Now, let's calculate the change in distance by subtracting s(π/3) from s(7π/3).
Change in distance = s(7π/3) - s(π/3)
= [(8√3 - 3)/2] - [(8√3 + 3)/2]
= (8√3 - 3 - 8√3 - 3)/2
= (-6)/2
= -3
The change in time is the difference between t = 7π/3 and t = π/3.
Change in time = 7π/3 - π/3
= 6π/3
= 2π
Finally, we can calculate the average velocity by dividing the change in distance by the change in time.
Average velocity = Change in distance / Change in time
= -3 / 2π
= -3/(2π)
Therefore, the average velocity of the particle from t = π/3 to t = 7π/3 is -3/(2π) centimeters per second.