Question

Zak jogs from his house to a lake. He then jogs 5 laps around the lake at a constant speed. The table shows thetotal amount of time, in minutes, Zak has been jogging after completing x laps.Which equation represents the time, m, in minutes, Zak has been jogging after completing x laps?A m = 12x + 5B m = 5x + 12C m = 24x + 5D m = 5x + 24

Answer 1

We are given Zak's laps jogged along with the minutes elapsed.

If the equation of a line is:

m = kx + b

Where m is the number of minutes.

k is the slope of the line

x is the Laps

b is the y-intercept (or where the line crosses the y-axis)

In order to get the equation of the relationship between Laps (x) and Minutes (m),

we need to calculate the slope k and intercept b.

The formulas for doing these are given below:

`\begin{gathered} m=(\sum(x_i-X)(y_i-Y))/((x_i-X)^2) \\ \text{where,} \\ x_i=\text{data points of Laps x} \\ X=\text{ Average of the Laps x} \\ y_i=\text{data points of Minutes m} \\ Y=\text{Average of Minutes m} \end{gathered}`

The formula for intercept (b) is;

`b=Y-kX`

where Y and X are the averages of m and x values from the table.

`\begin{gathered} Y=(\sum m)/(n)\text{ (n is the number of data values of Y)} \\ Y=(17+41+65)/(3) \\ \\ Y=41 \\ \\ X=(\sum x)/(n)\text{ (n is the number of data values of X)} \\ X=(1+3+5)/(3) \\ \\ X=3 \end{gathered}`

In order to be tidy and quick, a table is used to solve.

This table is shown in the image below:

Therefore, we can now calculate slope (m):

`\begin{gathered} m=(\sum(x_i-X)(y_i-Y))/((x_i-X)^2) \\ \\ m=(\mleft(-24\mright)\mleft(-2\mright)+0\mleft(0\mright)+\mleft(24\mright)\mleft(2\mright))/(4+0+4) \\ m=(96)/(8) \\ \\ m=12 \end{gathered}`

Now that we have slope (k) = 12, we can get the intercept b

`\begin{gathered} b=Y-kX \\ Y=41-12(3) \\ Y=41-36 \\ Y=5 \end{gathered}`

Therefore, the equation is:

m = 12x + 5