Answered

The table below shows the relationship between the approximate number of hours and the corresponding number of days on Saturn.




\multicolumn{2}{|c|}{Saturn}

Hours		Days

$266 \frac{2}{3}$		25

$373 \frac{1}{3}$		35

$426 \frac{2}{3}$		40

$586 \frac{2}{3}$		55






What is the approximate number of hours in 72 days on Saturn?

Verified Answer :

  • The relationship between hours and days on Saturn is linear: \(h = k \times d\).
  • Using the provided data, we find the constant ratio \(k = \frac{32}{3}\).
  • Applying this ratio for \(d=72\) days: \(h = \frac{32}{3} \times 72 = 768\).
  • The final answer is \(\boxed{768 \text{ hours}}\).

Explanation

  1. Data Analysis and Establishing the Relationship

To determine the relationship between hours and days on Saturn, we analyze the provided data points. Let \(h\) represent the number of hours and \(d\) represent the number of days. The data points are \((266\frac{2}{3}, 25)\), \((373\frac{1}{3}, 35)\), \((426\frac{2}{3}, 40)\), and \((586\frac{2}{3}, 55)\). We assume a linear relationship of the form \(h = k \times d\), where \(k\) is the constant ratio of hours per day. To verify this, we calculate the ratio \(k = \frac{h}{d}\) for each pair:
\begin{itemize}
\item For the first point: \(k = \frac{266\frac{2}{3}}{25} = \frac{800/3}{25} = \frac{800}{75} = \frac{32}{3}\).
\item For the second point: \(k = \frac{373\frac{1}{3}}{35} = \frac{1120/3}{35} = \frac{1120}{105} = \frac{32}{3}\).
\item For the third point: \(k = \frac{426\frac{2}{3}}{40} = \frac{1280/3}{40} = \frac{1280}{120} = \frac{32}{3}\).
\item For the fourth point: \(k = \frac{586\frac{2}{3}}{55} = \frac{1760/3}{55} = \frac{1760}{165} = \frac{32}{3}\).
\end{itemize}
Since the ratio is consistent across all data points, we have confirmed that the relationship is linear with \(k = \frac{32}{3}\).

  1. Calculating Hours for 72 Days

Now that we have established the constant ratio \(k = \frac{32}{3}\), we can use the formula \(h = k \times d\) to find the number of hours for \(d = 72\) days. Substituting our values:
\[h = \left(\frac{32}{3}\right) \times 72\]
\[h = 32 \times \left(\frac{72}{3}\right)\]
\[h = 32 \times 24\]
\[h = 768\]
Thus, there are \(768\) hours in \(72\) days on Saturn.

Examples

Understanding proportional relationships is essential for converting units, whether you are calculating travel time in different time zones, scaling up recipes, or adjusting map distances. In this case, by finding the constant ratio of hours to days, you can easily determine the equivalent time for any duration, which helps in planning schedules and understanding planetary cycles. This simple application of direct proportion illustrates how mathematical ratios simplify complex conversions.