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6. Howard decided to start jogging every day at the track. He completes the following chart for the month (4 weeks) that he jogs every day. Use the table to write an equation to represent the number of laps Howard runs if [tex]t[/tex] is the time in weeks since he began jogging.

\begin{tabular}{|l|l|l|l|l|}
\hline Time (weeks) & 0 & 1 & 2 & 3 \\
\hline Laps & 4 & 5 & 6 & 7 \\
\hline
\end{tabular}


Sagot :

Let's find an equation that represents the number of laps [tex]\( L \)[/tex] Howard runs if [tex]\( t \)[/tex] is the time in weeks since he began jogging.

1. Identify the given data:
- [tex]\( t \)[/tex] represents time in weeks.
- [tex]\( L \)[/tex] represents the number of laps.
- From the table, we have the following pairs [tex]\((t, L)\)[/tex]:
- [tex]\((0, 4)\)[/tex]
- [tex]\((1, 5)\)[/tex]
- [tex]\((2, 6)\)[/tex]
- [tex]\((3, 7)\)[/tex]

2. Assume a linear relationship:
- A linear relationship can be represented by the equation [tex]\( L = m \cdot t + c \)[/tex], where [tex]\( m \)[/tex] is the slope (rate of change) and [tex]\( c \)[/tex] is the y-intercept (initial value).

3. Determine the y-intercept [tex]\( c \)[/tex]:
- When [tex]\( t = 0 \)[/tex], [tex]\( L = 4 \)[/tex]. Thus, [tex]\( c = 4 \)[/tex].

4. Determine the slope [tex]\( m \)[/tex]:
- The slope [tex]\( m \)[/tex] represents the change in the number of laps per week. We can calculate it using the points:
- From week 0 to week 1:
[tex]\[ m = \frac{L_1 - L_0}{t_1 - t_0} = \frac{5 - 4}{1 - 0} = 1 \][/tex]
- Check for consistency using other points:
- From week 1 to week 2:
[tex]\[ m = \frac{6 - 5}{2 - 1} = 1 \][/tex]
- From week 2 to week 3:
[tex]\[ m = \frac{7 - 6}{3 - 2} = 1 \][/tex]
- The slope is consistently [tex]\( m = 1 \)[/tex].

5. Write the equation:
- Substitute the values of [tex]\( m \)[/tex] and [tex]\( c \)[/tex] into the linear equation:
[tex]\[ L = 1 \cdot t + 4 \][/tex]

Conclusively, the equation that represents the number of laps Howard runs if [tex]\( t \)[/tex] is the time in weeks since he began jogging is:
[tex]\[ L = t + 4 \][/tex]
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