To determine the slope of the line that represents the conversion between degrees and gradients, we follow the method of finding the slope between two points on the line.
Given the table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Degrees} & \text{Gradients} \\
\hline
-180 & -200 \\
\hline
-90 & -100 \\
\hline
0 & 0 \\
\hline
90 & 100 \\
\hline
180 & 200 \\
\hline
270 & 300 \\
\hline
\end{array}
\][/tex]
We choose two points to calculate the slope. For simplicity, let's pick the first and the last pairs:
- [tex]\((-180, -200)\)[/tex]
- [tex]\((180, 200)\)[/tex]
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\((x_1, y_1) = (-180, -200)\)[/tex] and [tex]\((x_2, y_2) = (180, 200)\)[/tex].
Substitute the values into the formula:
[tex]\[
m = \frac{200 - (-200)}{180 - (-180)}
\][/tex]
First, simplify the numerator and the denominator separately:
[tex]\[
200 - (-200) = 200 + 200 = 400
\][/tex]
[tex]\[
180 - (-180) = 180 + 180 = 360
\][/tex]
Now, put these values back into the slope formula:
[tex]\[
m = \frac{400}{360}
\][/tex]
Simplify the fraction:
[tex]\[
m = \frac{400}{360} = \frac{40}{36} = \frac{10}{9} \approx 1.1111111111111112
\][/tex]
Next, we round 1.1111111111111112 to the nearest hundredth:
[tex]\[
1.11
\][/tex]
Thus, the slope of the line representing the conversion of degrees to gradients is [tex]\(1.11\)[/tex].
