A stone is launched vertically upward from the top of a building with initial velocity $v_0=15$ $\text{m/s}$ and initial position $s_0=45$ $\text{m}$. Find the position and velocity functions for all times when the stone is above ground level ($s=0$). Use $g=9.8{\text{ m/s}}^2$.

A forest is subject to logging modeled by $\frac{ds}{dt}=0.06s-600$. For what values of $s_0$ is the forest shrinking? For what value of $s_0$ is the forest size constant? If $s_0=9,000$, when does the forest disappear?

Given the differential equation $z^{\prime }\left(t\right)+4z\left(t\right)=0$, how many arbitrary constants are present in its general solution?

A glass of hot tea is taken out of the kettle at a temperature of $90°\text{C}$ and left to cool in a room where the temperature is $20°\text{C}$. After $10$ minutes, the tea has cooled to $70°\text{C}$. At what time (after being taken out of the kettle) will the tea cool to $50°\text{C}$? Round your answer to the nearest minute.

Solve the initial value problem: $x^{\prime }\left(t\right)=3e^{2t}$, $x\left(0\right)=5$.