To solve this problem, we need to find the time taken and the distance traveled during the commute back home, given the different rates and time provided.
Let's break down the problem step-by-step:
1. Identify Given Information:
- Commute to Work:
- Rate: 55 miles per hour (mi/h)
- Time: [tex]\( t \)[/tex] hours
- Distance: [tex]\( 55t \)[/tex] miles (since Distance = Rate × Time)
- Commute to Home:
- Rate: 40 miles per hour (mi/h)
- Time: We need to determine it.
- Distance: ?
We can set up an equation based on the fact that the total round trip (to work and back home) should cover the same distance. Therefore, the distance covered going to work should be equal to the distance covered coming back home.
2. Distance Calculation:
From the table, the time for the commute to work is [tex]\( t \)[/tex].
When coming back home, we can assume that the time taken can be represented as [tex]\( (total\ time) - t \)[/tex].
If we consider the total time for the round trip and denote it as [tex]\( T \)[/tex], then the time taken for the return journey can be written as [tex]\( T - t \)[/tex].
However, the problem gives a hint that the time taken for returning home in hours is given to be “[tex]\( 1.25 - t \)[/tex]”. So, we assume that the given value [tex]\( 1.25 \)[/tex] represents the total time for both the commutes.
Thus, the time for commuting back home:
[tex]\[ 1.25 - t \, \text{hours} \][/tex]
3. Calculate Distance for Commute to Home:
We now use the formula Distance = Rate × Time for the commute back:
[tex]\[ \text{Distance home} = 40 \times (1.25 - t) \][/tex]
Therefore, we find:
[tex]\[ \text{Distance home} = 40 \times (1.25 - t) \, \text{miles} \][/tex]
So, the distance for the commute home is [tex]\( 40(1.25 - t) \)[/tex].
### Conclusion:
Now we have the missing values for the table:
\begin{tabular}{|c|c|c|c|}
\hline & \begin{tabular}{c} Rate \\ (mi/h) \end{tabular} & \begin{tabular}{c} Time \\ (h) \end{tabular} & \begin{tabular}{c} Distance \\ (miles) \end{tabular} \\
\hline \begin{tabular}{c} Commute to \\ Work \end{tabular} & 55 & [tex]$t$[/tex] & [tex]$55 t$[/tex] \\
\hline \begin{tabular}{c} Commute to \\ Home \end{tabular} & 40 & [tex]\(1.25 - t\)[/tex] & [tex]\(40(1.25 - t)\)[/tex] \\
\hline
\end{tabular}
These calculated values complete the table, clearly showing the rates, times, and distances for both commutes.
