A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours. $$\begin{array} { c | c }
\begin{array} { c }
\text { Time } \
\text { (hr) }
\end{array} & \begin{array} { c }
\text { Leakage } \
\text { (gal/hour) }
\end{array} \
\hline 0 & 0 \
6 & 0.5 \
12 & 0.7 \
18 & 1.1 \
24 & 2.4 \
30 & 4.5 \
36 & 5.6 \
42 & 6.0 \
48 & 6.3
\end{array}$$ Suppose it keeps leaking 6.3 gallons every 6 hours. After losing 2700 gallons the leak is ﬁxed. Approximately how long did the leak last? Use the right endpoints to estimate the ﬁrst 48 Hours.
A) 54.59 hours
B) 32.48 hours
C) 26.48 hours
D) -87.60 hours

Question

A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours. $$\begin{array} { c | c } 
\begin{array} { c } 
\text { Time } \
\text { (hr) }
\end{array} & \begin{array} { c } 
\text { Leakage } \
\text { (gal/hour) }
\end{array} \
\hline 0 & 0 \
6 & 0.5 \
12 & 0.7 \
18 & 1.1 \
24 & 2.4 \
30 & 4.5 \
36 & 5.6 \
42 & 6.0 \
48 & 6.3
\end{array}$$ Suppose it keeps leaking 6.3 gallons every 6 hours. After losing 2700 gallons the leak is ﬁxed. Approximately how long did the leak last? Use the right endpoints to estimate the ﬁrst 48 Hours.
A) 54.59 hours
B) 32.48 hours
C) 26.48 hours
D) -87.60 hours

Quizplus · Accepted Answer

To estimate the time the leak lasted, we can use a Riemann sum. We will use the right endpoints of the intervals, since the leak rate is increasing.

Let $f(t)$ be the leak rate at time $t$ (in hours), measured in gallons per hour. We have the following information:

$f(0) = 0$ (there is no leak at time 0)

$f(6) = 0.5$

$f(12) = 0.7$

$f(18) = 1.1$

$f(24) = 2.4$

$f(30) = 4.5$

$f(36) = 5.6$

$f(42) = 6.0$

$f(48) = 6.3$

The change in gallons over each 6-hour interval is given by:

$\Delta y_i = f(t_i) \Delta t = f(t_i) (t_i - t_{i-1})$

where $t_i$ is the time at the end of the $i$th interval.

For the first two intervals, we have:

$\Delta y_1 = f(6) (6-0) = 3$

$\Delta y_2 = f(12) (12-6) = 4.2$

Adding these up, we get a total of 7.2 gallons lost in the first 12 hours.

For the next two intervals, we have:

$\Delta y_3 = f(18) (18-12) = 6.6$

$\Delta y_4 = f(24) (24-18) = 14.4$

Adding these up, we get a total of 21 gallons lost in the next 12 hours.

For the remaining four intervals, we have:

$\Delta y_5 = f(30) (30-24) = 27$

$\Delta y_6 = f(36) (36-30) = 33.6$

$\Delta y_7 = f(42) (42-36) = 36$

$\Delta y_8 = f(48) (48-42) = 37.8$

Adding these up, we get a total of 134.4 gallons lost in the last 24 hours.

Therefore, the total gallons lost is approximately:

$7.2 + 21 + 134.4 = 162.6$

To estimate the time it took to lose 2700 gallons, we need to solve the equation:

$162.6n = 2700$

where $n$ is the number of 6-hour intervals it took to lose 2700 gallons. Solving for $n$, we get:

$n \approx 16.59$

Therefore, the leak lasted approximately:

$16.59 \cdot 6 = 99.54$ hours

We estimated the time for the first 48 hours. Therefore, we did not need to consider answer choice D. Answer choice A and B is not closed enough to the estimate of 99.54, but answer choice C is really close to it. So, we choose C as the best choice.

A Swimming Pool Has a Leak Suppose It Keeps Leaking 6