1. **Stating the problem:** We want to find the valid formula for the value $y$ at time $t$ during the interval $0 \leq t \leq T$, given that $y$ starts at $y_s$ and decreases to $y_T$ by time $T$. 2. **Understanding the problem:** The value $y$ decreases linearly from $y_s$ to $y_T$ as $t$ goes from 0 to $T$. This means $y$ changes at a constant rate. 3. **Formula for linear change:** The general formula for a linear change from $y_s$ to $y_T$ over time $T$ is: $$ y = y_s + \frac{y_T - y_s}{T} \cdot t $$ This formula starts at $y_s$ when $t=0$ and ends at $y_T$ when $t=T$. 4. **Rearranging the formula:** We can rewrite it as: $$ y = y_s - \frac{y_s - y_T}{T} \cdot t $$ This matches the fourth option. 5. **Checking other options:** - $y = y_s - y_T \cdot \frac{t}{T}$ is incorrect because it subtracts a fraction of $y_T$ instead of the difference $y_s - y_T$. - $y = y_s - y_T \cdot t$ is incorrect because it does not normalize by $T$ and will not reach $y_T$ at $t=T$. - $y = (y_s - y_T) \cdot \frac{t}{T}$ is incorrect because it starts at 0 when $t=0$ instead of $y_s$. 6. **Final answer:** The valid relation is: $$ y = y_s - \frac{y_s - y_T}{T} \cdot t $$