The displacement of a particle is described by `x = t + 2t^(2) +3t^(3) -4t^(4)` The acceleration at t = 3s is

Options

A. + 378 units
B. `+ 374` units
C. `- 374` units
D. `-378` units

Explanation

To find the acceleration of the particle at \( t = 3 \) seconds, we will follow these steps:

### Step 1: Write down the displacement equation
The displacement of the particle is given by:
\[ x(t) = t + 2t^2 + 3t^3 - 4t^4 \]

### Step 2: Find the velocity
To find the velocity, we need to take the first derivative of the displacement with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(t + 2t^2 + 3t^3 - 4t^4) \]
Calculating the derivative:
\[ v(t) = 1 + 4t + 9t^2 - 16t^3 \]

### Step 3: Find the acceleration
Next, we find the acceleration by taking the derivative of the velocity:
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(1 + 4t + 9t^2 - 16t^3) \]
Calculating the derivative:
\[ a(t) = 0 + 4 + 18t - 48t^2 \]
Thus, the acceleration equation simplifies to:
\[ a(t) = 4 + 18t - 48t^2 \]

### Step 4: Substitute \( t = 3 \) seconds into the acceleration equation
Now we substitute \( t = 3 \) into the acceleration equation:
\[ a(3) = 4 + 18(3) - 48(3^2) \]
Calculating each term:
\[ a(3) = 4 + 54 - 48(9) \]
\[ a(3) = 4 + 54 - 432 \]
\[ a(3) = 58 - 432 \]
\[ a(3) = -374 \]

### Final Answer
The acceleration at \( t = 3 \) seconds is:
\[ \boxed{-374 \text{ units}} \]

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