1. **Problem statement:** We want to find the equation for the rocket weight $W$ at time $t$ where $0 \leq t \leq T$, given initial weight $W_1$ and final weight $W_T$ at time $T$. 2. **Understanding the problem:** The weight decreases linearly because fuel is expelled proportionally to time. So, $W$ decreases from $W_1$ at $t=0$ to $W_T$ at $t=T$. 3. **General linear equation:** A linear function from $(0, W_1)$ to $(T, W_T)$ is given by $$ W = W_1 + m t $$ where $m$ is the slope. 4. **Calculate slope $m$:** $$ m = \frac{W_T - W_1}{T - 0} = \frac{W_T - W_1}{T} $$ 5. **Substitute slope into equation:** $$ W = W_1 + \frac{W_T - W_1}{T} t $$ 6. **Rewrite to match options:** Since $W_T < W_1$ (weight decreases), rewrite as $$ W = W_1 - \frac{W_1 - W_T}{T} t $$ 7. **Check options:** The correct formula is $$ W = W_1 - \left(\frac{W_1 - W_T}{T}\right) t $$ which matches the bottom-right box. **Final answer:** $$ \boxed{W = W_1 - \left(\frac{W_1 - W_T}{T}\right) t} $$