Question

1. explain the meaning of the point (0, 0). 9. how long does it take each horse to run a mile? 10. multiple representations write an equation for the relationship between time and distance for each horse. 11. draw conclusions at the given rates, how far would each horse run in 12 minutes? 12. analyze relationships draw a line on the graph representing a horse than runs faster than horses a and b.

Explanation:

Step1: Find the speed of each horse

For horse A, when $t = 10$ min, $d= 3$ mi. Speed $v_A=\frac{d}{t}=\frac{3}{10}$ mi/min. To find the time to run 1 mile, use $t=\frac{d}{v}$, so $t_A=\frac{1}{\frac{3}{10}}=\frac{10}{3}\approx3.33$ min. For horse B, when $t = 10$ min, $d = 5$ mi. Speed $v_B=\frac{d}{t}=\frac{5}{10}=\frac{1}{2}$ mi/min. Time to run 1 mile $t_B=\frac{1}{\frac{1}{2}} = 2$ min.

Step2: Write the equations

The general equation for a linear - relationship between distance $d$ and time $t$ is $d=vt$. For horse A, $d_A=\frac{3}{10}t$. For horse B, $d_B=\frac{1}{2}t$.

Step3: Find the distance in 12 minutes

For horse A, substitute $t = 12$ into $d_A=\frac{3}{10}t$, so $d_A=\frac{3}{10}\times12=\frac{36}{10}=3.6$ mi. For horse B, substitute $t = 12$ into $d_B=\frac{1}{2}t$, so $d_B=\frac{1}{2}\times12 = 6$ mi.

Step4: Draw a faster - horse line

A faster horse has a steeper slope. A line with a slope greater than $\frac{1}{2}$ (the slope of horse B) can be drawn. For example, a line with slope $1$ passing through the origin $d=t$ would be steeper than both A and B.

Answer:

1. Horse A: $\frac{10}{3}$ min, Horse B: 2 min
2. Horse A: $d=\frac{3}{10}t$, Horse B: $d=\frac{1}{2}t$
3. Horse A: 3.6 mi, Horse B: 6 mi
4. Draw a line with a slope greater than $\frac{1}{2}$ passing through the origin.