\( a(t) \) is the acceleration, \( v(t) \) is the velocity, and \( s(t) \) is the position. Find the missing information. \[ \begin{array}{l}a(t)=2 t \\ v(0)=7 \\ s(3)=31 \\ s(1)=\frac{[?]}{[\square]}\end{array} \]

To find \( s(1) \), we first need to determine the velocity \( v(t) \) by integrating the acceleration \( a(t) = 2t \): \[ v(t) = \int a(t) \, dt = \int 2t \, dt = t^2 + C \] Using the initial condition \( v(0) = 7 \), we can find the constant \( C \): \[ v(0) = 0^2 + C = 7 \implies C = 7 \] Thus, the velocity function is: \[ v(t) = t^2 + 7 \] Next, we find the position \( s(t) \) by integrating the velocity \( v(t) \): \[ s(t) = \int v(t) \, dt = \int (t^2 + 7) \, dt = \frac{t^3}{3} + 7t + D \] To find the constant \( D \), we use the given condition \( s(3) = 31 \): \[ s(3) = \frac{3^3}{3} + 7 \cdot 3 + D = 31 \] \[ s(3) = 9 + 21 + D = 31 \implies D = 31 - 30 = 1 \] Now, the position function is: \[ s(t) = \frac{t^3}{3} + 7t + 1 \] Finally, we compute \( s(1) \): \[ s(1) = \frac{1^3}{3} + 7 \cdot 1 + 1 = \frac{1}{3} + 7 + 1 = \frac{1}{3} + 8 = \frac{1 + 24}{3} = \frac{25}{3} \] Thus, the missing information is: \[ s(1) = \frac{25}{3} \]