4. Rosa watched the distance markers along the highway. She noticed that she had traveled $$ 3 \mathrm{~km} \mathrm{in} 2 \mathrm{~min}, 6 \mathrm{~km} $$ in 4 min , and 9 km in 6 min . Does this graph show direct variation? If so, give the constant of proportionality. How far will she have traveled in 10 min ?

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UpStudy AI · Accepted Answer

1. Rosa's travel measurements are given as:
   - $$ 3 $$ km in $$ 2 $$ minutes,
   - $$ 6 $$ km in $$ 4 $$ minutes, and
   - $$ 9 $$ km in $$ 6 $$ minutes.

2. To determine if the graph shows direct variation, we need to check if the ratio of distance to time is constant. We calculate the rate (slope) for each measurement:

$$
   \text{Rate} = \frac{\text{Distance}}{\text{Time}}
   $$

- For $$ 3 $$ km in $$ 2 $$ minutes:
     
     $$
     \frac{3}{2} = 1.5 \, \text{km/min}
     $$

- For $$ 6 $$ km in $$ 4 $$ minutes:
     
     $$
     \frac{6}{4} = 1.5 \, \text{km/min}
     $$

- For $$ 9 $$ km in $$ 6 $$ minutes:
     
     $$
     \frac{9}{6} = 1.5 \, \text{km/min}
     $$

3. Since the rate is constant at $$ \frac{3}{2} \, \text{km/min} $$, the graph shows direct variation with the constant of proportionality $$ k = \frac{3}{2} $$.

4. To find the distance Rosa has traveled in $$ 10 $$ minutes, use the direct variation formula:

$$
   \text{Distance} = k \times \text{Time}
   $$

Substitute $$ k = \frac{3}{2} $$ and $$ \text{Time} = 10 $$ minutes:

$$
   \text{Distance} = \frac{3}{2} \times 10 = 15 \, \text{km}
   $$

Thus, the graph shows direct variation with a constant of proportionality $$ \frac{3}{2} $$, and Rosa will have traveled $$ 15 $$ km in $$ 10 $$ minutes.
Rosa's travel shows direct variation with a constant of proportionality of $$ \frac{3}{2} $$ km/min. In 10 minutes, she will have traveled 15 km.

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