Estimate the value of the quantity.
-A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours.
$\begin{array}{c|c}
\begin{array}{c}
\text { Time } \
\text { (hr) }
\end{array} & \begin{array}{c}
\text { Leakage } \
\text { (gal/hour) }
\end{array} \
\hline 0 & 0 \
6 & 0.5 \
12 & 0.7 \
18 & 1.1 \
24 & 2.4 \
30 & 4.5 \
36 & 5.6 \
42 & 6.0 \
48 & 6.7
\end{array}$

Suppose it keeps leaking $6.7$ gallons every 6 hours. After losing 3200 gallons the leak is fixed. Approximately how long did the leak last? Use the right endpoints to estimate the first 48 hours.
A) 34.15 hours
B) 54.88 hours
C) 28.15 hours
D) -85.00 hours

Question

Estimate the value of the quantity.
-A swimming pool has a leak. The leak is getting worse. The following table gives the leak rate every 6 hours.
$\begin{array}{c|c}
\begin{array}{c}
\text { Time } \
\text { (hr) }
\end{array} & \begin{array}{c}
\text { Leakage } \
\text { (gal/hour) }
\end{array} \
\hline 0 & 0 \
6 & 0.5 \
12 & 0.7 \
18 & 1.1 \
24 & 2.4 \
30 & 4.5 \
36 & 5.6 \
42 & 6.0 \
48 & 6.7
\end{array}$

Suppose it keeps leaking $6.7$ gallons every 6 hours. After losing 3200 gallons the leak is fixed. Approximately how long did the leak last? Use the right endpoints to estimate the first 48 hours.
A) 34.15 hours
B) 54.88 hours
C) 28.15 hours
D) -85.00 hours

Quizplus · Accepted Answer

We can estimate the total leakage by adding the leakage for each 6-hour interval. Using the right endpoint of each interval to estimate the leakage, we have:

\begin{align*}
	ext{Total leakage} &= (0.5)(6) + (0.7)(6) + (1.1)(6) + (2.4)(6)\
&\quad + (4.5)(6) + (5.6)(6) + (6.0)(6) + (6.7)(6)\
&= 314.1	ext{ gallons}
\end{align*}

Since the leak is fixed after losing 3200 gallons, we can estimate the duration of the leak by dividing the total leakage by the rate of leakage in the last 6-hour interval:

\begin{align*}
	ext{Duration of leak} &= \frac{3200	ext{ gallons}}{6.7	ext{ gallons/hour}}\
&\approx 477.61	ext{ hours}
\end{align*}

However, we only want to estimate the duration of the leak during the first 48 hours, so we need to adjust our calculation. Using the right endpoint of the final interval that falls within the first 48 hours (which is the 6-hour interval from 42 to 48 hours), we have:

\begin{align*}
	ext{Adjusted total leakage} &= (0.5)(6) + (0.7)(6) + (1.1)(6) + (2.4)(6)\
&\quad + (4.5)(6) + (5.6)(6) + (6.0)(6)\
&= 294.6	ext{ gallons}
\end{align*}

The amount of leakage within the first 48 hours is then:

\begin{align*}
	ext{Leakage within 48 hours} &= 3200	ext{ gallons} - 	ext{Adjusted total leakage}\
&= 3200	ext{ gallons} - 294.6	ext{ gallons}\
&= 2905.4	ext{ gallons}
\end{align*}

We can estimate the duration of the leak during the first 48 hours by dividing the amount of leakage within the first 48 hours by the rate of leakage in the last 6-hour interval:

\begin{align*}
	ext{Duration of leak within first 48 hours} &= \frac{2905.4	ext{ gallons}}{6.7	ext{ gallons/hour}}\
&\approx 432.69	ext{ hours}
\end{align*}

Therefore, the best choice is C, approximately 28.15 hours.

Estimate the Value of the Quantity Suppose It Keeps Leaking 6.76.76.7

Estimate the Value of the Quantity Suppose It Keeps Leaking $6.7$