Question

A machine at a bottling company fills water bottles. the number of bottles filled is proportional to the amount of time the machine runs. A worker records how many bottles are filled on four consecutive days and how many hours the machine runs on those days. The worker's data are shown in the table.

Answer 1

In the table shown below:

Let N be the number of bottles filled,

Let T be the time in hours.

Given that the number of bottles filled is proportional to the amount of time the machine runs, we have

`\begin{gathered} N\propto T \\ \text{Introducing a proportionality constant, we have } \\ N\text{ = kT} \\ \Rightarrow k\text{ = }(N)/(T) \\ \end{gathered}`

Let's evaluate the value of k for each day.

Thus, on monday,

`\begin{gathered} N\text{ = 9900} \\ T\text{ = 5.5} \\ \text{thus,} \\ k\text{ = }(9900)/(5.5) \\ \Rightarrow k\text{ = 1800} \end{gathered}`

Tuesday:

`\begin{gathered} N\text{ = }11160 \\ T\text{ = }6.2 \\ \text{thus,} \\ k\text{ = }(11160)/(6.2) \\ \Rightarrow k\text{ = 1800} \end{gathered}`

Wednesday:

`\begin{gathered} N\text{ = }12330 \\ T\text{ = }6.25 \\ \text{thus,} \\ k\text{ = }(12330)/(6.25) \\ \Rightarrow k\text{ = 1972.8} \end{gathered}`

Thursday:

`\begin{gathered} N\text{ = }10440 \\ T\text{ = }5.80 \\ \text{thus,} \\ k\text{ = }(10440)/(5.8) \\ \Rightarrow k=1800 \end{gathered}`

It is observed that all exept wednesday have the same value of k.

Thus, the amount of time required for the number of bottles filled on wednesday is evaluated as

`\begin{gathered} N\text{ = }12330 \\ k\text{ = 1800} \\ T\text{ = ?} \\ \text{but} \\ k\text{ = }(N)/(T) \\ 1800\text{ = }(12330)/(T) \\ \Rightarrow T\text{ = }\frac{\text{12330}}{1800} \\ T\text{ = 6.85} \end{gathered}`

Hence, the incorrect day is Wednesday. The amount of time for that many bottles should be 6.85 hours.