Question

3
a rabbit and turtle are in a race. is the relationship between distance traveled and time proportional for either one? if so, write an equation that represents the relationship.
turtles run:

distance (meters)	108	405	540	1,768.5

rabbits run:

distance (meters)	800	900	1,107.5	1,524

Explanation:

Step1: Check turtle's proportionality

Calculate $\frac{\text{Distance}}{\text{Time}}$ for each pair:
$\frac{108}{2}=54$, $\frac{405}{7.5}=54$, $\frac{540}{10}=54$, $\frac{1768.5}{32.75}=54$

Step2: Turtle's equation setup

Since the ratio is constant ($54$ m/min), distance $d$ is proportional to time $t$:
$d=54t$

Step3: Check rabbit's proportionality

Calculate $\frac{\text{Distance}}{\text{Time}}$ for each pair:
$\frac{800}{1}=800$, $\frac{900}{5}=180$, $\frac{1107.5}{20}=55.375$, $\frac{1524}{32.5}=46.89$

Step4: Rabbit's proportionality conclusion

The ratios are not equal, so the relationship is not proportional.

Answer:

1. Turtle: The relationship is proportional; $d=54t$
2. Rabbit: The relationship is not proportional