The acceleration of a particle moving along the x-axis at time t is given by a(t)=6t-2. If the position is 10 when 1 , then which of the following could be the function for position x(t)= ? (A) 9t^(2)+1 (B) 3t^(2)-2t+4 (IC) t^(3)-t^(2)+4t+6 (D) t^(3)-t^(2)+9t-20 (E) 36t^(3)-4t^(2)-8t+5

Question

The acceleration of a particle moving along the  x-axis at time  t is given by  a(t)=6t-2. If the position is 10 when 1 , then which of the following could be the function for position  x(t)= ? (A)  9t^(2)+1 (B)  3t^(2)-2t+4 (IC)  t^(3)-t^(2)+4t+6 (D)  t^(3)-t^(2)+9t-20 (E)  36t^(3)-4t^(2)-8t+5

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Integrate acceleration for velocity: To find the position function x(t), we need to integrate the acceleration function a(t) = 6t - 2. The integral of acceleration gives us the velocity function v(t).Find constant for velocity: The indefinite integral of a(t) with respect to t is: ∫(6t - 2) dt = 3t^2 - 2t + C, where C is the constant of integration for the velocity function.Integrate velocity for position: To find the constant C, we need an initial condition for velocity. Since we are not given the initial velocity, we cannot find the exact value of C at this step. We will proceed to find the position function x(t) and use the given initial position to find the constant for the position function.Find constant for position: Now we integrate the velocity function to get the position function x(t): ∫(3t^2 - 2t + C) dt = t^3 - t^2 + Ct + D, where D is the constant of integration for the position function.Use initial position to find constants: We are given that the position is 10 when t = 1. We use this condition to find D: x(1) = 1^3 - 1^2 + C*1 + D = 10 1 - 1 + C + D = 10 C + D = 10 Since we do not have the value of C from the velocity function, we cannot find the exact values of C and D. However, we can compare the structure of the resulting position function with the given options to see which one could potentially match.Compare position function with options: We compare the structure of the position function t^3 - t^2 + Ct + D with the given options: (A) 9t^(2)+1 does not match because it lacks the t^3 term. (B) 3t^(2)-2t+4 does not match because it lacks the t^3 term and has the wrong coefficient for the t^2 term. (C) t^(3)-t^(2)+4t+6 has the correct t^3 and t^2 terms, but we cannot confirm the coefficients for the t and constant terms without the initial velocity. (D) t^(3)-t^(2)+9t-20 has the correct t^3 and t^2 terms, but again, we cannot confirm the coefficients for the t and constant terms without the initial velocity. (E) 36t^(3)-4t^(2)-8t+5 does not match because it has incorrect coefficients for the t^3 and t^2 terms.Analyze options for correct function: Since options (C) and (D) have the correct t^3 and t^2 terms, we need to determine which one could be the correct position function based on the given initial position x(1) = 10. For option (C): x(1) = 1^3 - 1^2 + 4*1 + 6 = 1 - 1 + 4 + 6 = 10. This matches the given initial condition. For option (D): x(1) = 1^3 - 1^2 + 9*1 - 20 = 1 - 1 + 9 - 20 = -11. This does not match the given initial condition.Therefore, the correct function for the position x(t) is option (C) t^(3) - t^(2) + 4t + 6, as it matches the structure of the position function we derived and satisfies the initial condition x(1) = 10.

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