A mass hanging from a spring is set in motion and its ensuing velocity is given by \( v(t)=-3 \pi \sin \pi t \) for \( t \geq 0 \). Assume that the positive direction is upward and \( s(0)=3 \). a. Determine the position function for \( t \geq 0 \). b. Graph the position function on the interval \( [0,3] \). c. At what times does the mass reach its lowest point the first three times? d. At what times does the mass reach its highest point the first three times?

**Problem a. Determine the position function for \( t \geq 0 \).** To find the position function \( s(t) \) of the mass, we'll start by integrating the given velocity function \( v(t) \). Given: \[ v(t) = \frac{ds}{dt} = -3\pi \sin(\pi t) \] **Step 1: Integrate the velocity function to find the position function.** \[ s(t) = \int v(t) \, dt = \int -3\pi \sin(\pi t) \, dt \] **Step 2: Perform the integration.** \[ s(t) = -3\pi \int \sin(\pi t) \, dt \] Recall that: \[ \int \sin(\pi t) \, dt = -\frac{1}{\pi} \cos(\pi t) + C \] Substituting back: \[ s(t) = -3\pi \left( -\frac{1}{\pi} \cos(\pi t) \right) + C = 3\cos(\pi t) + C \] **Step 3: Use the initial condition to find the constant \( C \).** Given \( s(0) = 3 \): \[ s(0) = 3\cos(0) + C = 3(1) + C = 3 + C = 3 \] \[ \Rightarrow C = 0 \] **Final Position Function:** \[ s(t) = 3\cos(\pi t) \] **Answer:** a. The position function is \( s(t) = 3\, \cos(\pi t) \) for all \( t \geq 0 \).