1. **State the problem:** We want to find how many people left the auditorium between minutes $1$ and $5$ given the rate of change of people remaining is $r(t) = -0.1^t$ people per minute. 2. **Understand the rate function:** The function $r(t)$ represents the rate at which people are leaving (negative means decreasing number of people). 3. **Recall the integral interpretation:** The total change in the number of people between times $a$ and $b$ is given by the integral $$\int_a^b r(t)\,dt.$$ Since $r(t)$ is negative, the number of people decreases. 4. **Find the number of people who left:** The number of people who left is the positive amount of decrease, which is $$-\int_1^5 r(t)\,dt = \int_1^5 -r(t)\,dt.$$ This corresponds to option B. 5. **Check other options:** - Option A: $75 + \int_1^5 r(t)\,dt$ would give the number of people remaining at $t=5$, not the number who left. - Option C: $75 - \int_1^5 r(t)\,dt$ is incorrect because subtracting a negative integral would increase the number. - Option D: $\int r(t)\,dt$ is indefinite and does not specify limits. **Final answer:** The correct expression to find how many people left between minutes $1$ and $5$ is $$\int_1^5 -r(t)\,dt.$$