Final answer:
To find when the speed is 5, we differentiate the x-component of the position function r(t), find the magnitude of the resultant velocity, set it equal to 5, and solve for t. The point has a speed of 5 at t = 4.5 and t = -0.5.
Step-by-step explanation:
To find the values of t where the speed of the point is 5 for the position function r(t) = ((t - 2)², 2), we first need to calculate the velocity of the point. The velocity is the derivative of the position function.
The position function only has an x-component that changes with time, given by (t - 2)². The change in the y-component is zero since it is constant at 2. Hence, for the velocity vector v(t), the derivative of (t - 2)² with respect to t is 2(t - 2). Since the y-component is constant, its derivative is zero.
To find the speed, we calculate the magnitude of the velocity vector. The magnitude of v(t) is the square root of the sum of the squares of its components. Thus, the speed is |v(t)| = √[(2(t - 2))² + 0²], which simplifies to |v(t)| = 2|t - 2|. We set this equal to 5 to find the desired values of t.
Setting the speed equation to 5 yields 2|t - 2| = 5. Dividing both sides by 2 gives |t - 2| = 2.5. This results in two possible solutions, t - 2 = 2.5 and t - 2 = -2.5, leading to t values of 4.5 and -0.5, respectively. Therefore, the point has a speed of 5 at t = 4.5 and t = -0.5.