To solve this problem, we need to carefully analyze Joylin's steps to determine where she made her first error. Here are the details of her steps:
Step 1: Identify the mistake in calculating [tex]\( k \)[/tex]
Joylin is trying to model a proportional relationship between [tex]\( y \)[/tex] (the total cost in dollars) and [tex]\( x \)[/tex] (the number of videos downloaded). She knows that the cost to download 3 videos is $12.
1. To find the constant of proportionality [tex]\( k \)[/tex]:
[tex]\[ k = \text{ratio of total cost to number of videos} \][/tex]
2. The correct formula for [tex]\( k \)[/tex] should be:
[tex]\[ k = \frac{\text{total cost}}{\text{number of videos}} \][/tex]
3. Given the information:
[tex]\[ \text{Total cost} = 12 \text{ dollars} \][/tex]
[tex]\[ \text{Number of videos} = 3 \][/tex]
4. Therefore:
[tex]\[ k = \frac{12}{3} = 4 \][/tex]
However, Joylin's work shows:
[tex]\[ k = \frac{3}{12} = 0.25 \][/tex]
Step 2: Identifying the mistake:
1. Joylin calculated:
[tex]\[ k = \frac{3}{12} \][/tex]
Here, she mistakenly divided the number of videos (3) by the total cost (12), instead of dividing the total cost by the number of videos.
2. The correct calculation should be:
[tex]\[ k = \frac{12}{3} = 4 \][/tex]
3. The correct equation modeling the proportional relationship should be:
[tex]\[ y = 4x \][/tex]
Conclusion:
Joylin's first error was in switching the numerator and the denominator when she found [tex]\( k \)[/tex]. She should have used the total cost as the numerator and the number of videos as the denominator, rather than the other way around.
Therefore, the correct answer is:
Joylin switched the numerator and the denominator when she found [tex]\( k \)[/tex].
