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]
