Question

Write an equation for the line tangent to the path of the particle at time t = 3. Find the y-coordinate of the particle's position at time t = 0. Find the total distance traveled by the particle.

Answer 1

To write the equation for the tangent line at t = 3, find the slope using the endpoints of the tangent line. The y-coordinate of the particle's position at t = 0 is 4.0. Calculate the total distance traveled by integrating the absolute value of the velocity function over the given interval.

Step-by-step explanation:

To write an equation for the line tangent to the path of the particle at time t = 3, we need to find the slope of the tangent line. First, we determine the endpoints of the tangent line by finding the position of the particle at 19 s and 32 s. These correspond to a position of 1,300 m at 19 s and a position of 3,120 m at 32 s. Then, we use the endpoints to calculate the slope of the line. The slope of the tangent line is equal to the change in y-coordinates divided by the change in x-coordinates.

To find the y-coordinate of the particle's position at time t = 0, we need to evaluate the position function at t = 0. The given position function is x(t) = 4.0 + 2.0t. By plugging in t = 0, we get x(0) = 4.0 + 2.0(0) = 4.0. Therefore, the y-coordinate of the particle's position at t = 0 is 4.0.

To find the total distance traveled by the particle, we can calculate the area under the velocity-time graph. The velocity function is given as
`v(t) = 3.0 + 1.5t^2`. To find the distance traveled between t = 1.0 s and t = 3.0 s, we need to integrate the absolute value of the velocity function over that interval.