Final answer:
Kaley and Natalia are participating in a food collection challenge. Kaley starts with 14 cans, Natalia with 6. After 8 days, both will have collected 38 cans, after which Natalia will begin to collect more than Kaley due to her higher daily collection rate.
Step-by-step explanation:
Kaley and Natalia are participating in a challenge to see who can collect the most cans of food to donate. Kaley has already collected 14 cans and is collecting 3 more each day. Natalia has collected 6 cans and is collecting 4 more each day. To analyze who will collect more cans and when, we can define two linear equations based on their initial numbers of cans and their daily collection rates:
- Kaley's cans = 14 + 3d, where d represents days.
- Natalia's cans = 6 + 4d.
By setting the two equations equal to each other, we can determine when Natalia will catch up to Kaley or if Kaley will continue to lead.
14 + 3d = 6 + 4d
This can be solved as follows:
- Subtract 3d from both sides: 14 = 6 + d
- Subtract 6 from both sides: 8 = d
Thus, after 8 days, Natalia will have the same number of cans as Kaley. To find out who will have more after 8 days, we need to continue collecting: Kaley will have 14 + (3 × 8) = 38 cans, and Natalia will have 6 + (4 × 8) = 38 cans.
After this point, since Natalia's daily collection rate is higher, she will surpass Kaley in the number of cans collected.