Answer is: option4

3.696

Solution:

The position function is given by:

\[ x(t) = 3\sin t + t^2 + 7 \]

Step 1: Find the Velocity

Velocity is the first derivative of position:

\[ v(t) = \frac{dx}{dt} = 3\cos t + 2t \]

Step 2: Find the Acceleration

Acceleration is the first derivative of velocity:

\[ a(t) = \frac{dv}{dt} = -3\sin t + 2 \]

Step 3: Find when Acceleration is Zero

Setting acceleration to zero:

\[ -3\sin t + 2 = 0 \]

\[ \sin t = \frac{2}{3} \]

Solving for \( t \):

\[ t = \arcsin\left(\frac{2}{3}\right) \]

Using a calculator:

\[ t \approx 0.7297 \]

Step 4: Compute Velocity at \( t = 0.7297 \)

\[ v(0.7297) = 3\cos(0.7297) + 2(0.7297) \]

Approximating:

\[ \cos(0.7297) \approx 0.7453 \]

\[ v(0.7297) = 3(0.7453) + 2(0.7297) \]

\[ = 2.2359 + 1.4594 \]

\[ = 3.6953 \]

Final Answer: 3.696