Final answer:
The velocity of the particle at time t is given by the function v(t) = 3t² - 6t.
Step-by-step explanation:
To find the velocity of the particle at time t, we can take the derivative of the position function x(t). So, differentiate x(t) with respect to t:
v(t) = d/dt (t³ - 3t² - 9t)
v(t) = 3t² - 6t
The velocity of the particle at any time t is the rate of change of its position with respect to time. Mathematically, this is expressed as the derivative of the position function x(t) with respect to time t. The problem provides the position function: x(t) = t³ - 3t² - 9t To find the velocity function v(t), we take the derivative of x(t) with respect to t. Using the power rule for differentiation, the derivative of t^n with respect to t is n*t^(n-1), where n is a real number.
The constant coefficient rule states that the derivative of a constant times a function is the constant times the derivative of the function. Applying these rules, we differentiate each term of the position function: 1. The derivative of t³ with respect to t is 3*t^(3-1) = 3t². 2. The derivative of -3t² with respect to t is -3*2*t^(2-1) = -6t. 3. The derivative of -9t with respect to t is -9 (since the derivative of t is 1).
Therefore, the velocity function v(t) is the sum of these derived terms: v(t) = 3t² - 6t - 9 Comparing the resulting velocity function with the options given, we see that the correct answer is: A. 3t² - 6t - 9
Therefore, the velocity of the particle at time t is given by the function v(t) = 3t² - 6t.