Question

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:
rabbits run:
problem #6 - review from grade 7 (spicy)

Explanation:

Step1: Calculate the ratio for Turtle

For Turtle:

• When distance \(d = 108\) meters and time \(t=2\) minutes, the ratio \(\frac{d}{t}=\frac{108}{2} = 54\)
• When \(d = 405\) meters and \(t = 7.5\) minutes, \(\frac{d}{t}=\frac{405}{7.5}=54\)
• When \(d = 540\) meters and \(t = 10\) minutes, \(\frac{d}{t}=\frac{540}{10}=54\)
• When \(d = 1768.5\) meters and \(t = 32.75\) minutes, \(\frac{d}{t}=\frac{1768.5}{32.75}=54\)

Since the ratio \(\frac{d}{t}\) is constant (\(k = 54\)), the relationship for the Turtle is proportional. The equation for a proportional relationship is \(d=kt\), so for the Turtle \(d = 54t\)

Step2: Calculate the ratio for Rabbit

For Rabbit:

• When \(d = 800\) meters and \(t = 1\) minute, \(\frac{d}{t}=\frac{800}{1}=800\)
• When \(d = 900\) meters and \(t = 5\) minutes, \(\frac{d}{t}=\frac{900}{5}=180\)
• Since \(800

eq180\), the ratio \(\frac{d}{t}\) is not constant. So the relationship for the Rabbit is not proportional

Answer:

The relationship between distance and time is proportional for the Turtle. The equation is \(d = 54t\). The relationship is not proportional for the Rabbit.