Question

water leaks from a tank at a rate r( t ) where r( t ) = 2 + 0.144 t gallons per hour where t is the number of hours since 7 am.
interpret \\(\int_{3}^{7} (2 + 0.144t) dt = 10.88\\).
note that the \hours\ indicate the time of day in military time.
\\(\circ\\) none of the other answers are correct.
\\(\circ\\) between 2 and 7 hours, the tank lost 10.88 gallons.
\\(\circ\\) between 10 and 14 hours, the tank lowered by 10.88 feet.
\\(\circ\\) between 10 and 14 hours, the tank lost 10.88 gallons.
\\(\circ\\) between 3 and 7 hours, the tank lost 10.88 gallons.

Explanation:

1. First, map the values of $t$ to actual time: $t$ is hours since 7 AM. When $t=3$, the time is $7 + 3 = 10$ AM (10 hours military time). When $t=7$, the time is $7 + 7 = 14$ hours (2 PM, military time).
2. The integral of a rate function over an interval gives the total quantity corresponding to that rate over the interval. Here, $R(t)$ is the leak rate in gallons per hour, so the integral $\int_{3}^{7} R(t) dt$ gives the total gallons of water lost between $t=3$ and $t=7$.
3. Eliminate incorrect options:

• Option 1: Incorrect, as a valid option exists.
• Option 2: Incorrect, it uses $t$ values directly as time instead of converting from hours since 7 AM.
• Option 3: Incorrect, the unit is gallons, not feet.
• Option 5: Incorrect, it uses $t$ values directly as time instead of converting from hours since 7 AM.

Answer:

Between 10 and 14 hours, the tank lost 10.88 gallons.