A gardener measured the heights of two plants at the end of every week. The function y = 3x + 8.5 gives the height of Plant A in centimeters at the end of x weeks. The function y = 2.5x + 14.5 gives the height of Plant B in centimeters at the end of x weeks.
Based on this information, which of these statements is true?
Responses:
Plant A and Plant B will be the same height at the end of Week 3.
Plant A and PLant B will be the same height at the end of Week 12.
The height of Plant A will always be greater than the height of Plant B.
The height of Plant B will always be greater than the height of Plant A.
Let's analyze each of the statements:
1. Plant A and Plant B will be the same height at the end of Week 3.
To evaluate if this is true, plug in x=3 into both equations:
Plant A: y = 3(3) + 8.5 = 9 + 8.5 = 17.5 cm
Plant B: y = 2.5(3) + 14.5 = 7.5 + 14.5 = 22 cm
At the end of Week 3, Plant A is 17.5 cm tall, and Plant B is 22 cm tall. Thus, this statement is false.
2. Plant A and Plant B will be the same height at the end of Week 12.
To check if this statement is true, let's plug x=12 into both equations:
Plant A: y = 3(12) + 8.5 = 36 + 8.5 = 44.5 cm
Plant B: y = 2.5(12) + 14.5 = 30 + 14.5 = 44.5 cm
At the end of Week 12, Plant A is 44.5 cm tall, and Plant B is also 44.5 cm tall. So, this statement is true.
3. The height of Plant A will always be greater than the height of Plant B.
This cannot be true because we've already found a week (Week 12) where both plants are the same height. Also, by looking at the slopes of the two functions (3 for Plant A and 2.5 for Plant B), we can see that Plant A grows faster every week by 0.5 cm compared to Plant B. However, Plant B starts at a greater initial height (14.5 cm) than Plant A (8.5 cm). There may be some time in the future when Plant A surpasses Plant B, but the statement is not true for all x (all weeks).
4. The height of Plant B will always be greater than the height of Plant A.
This statement is false for the same reasons mentioned above. Plant B starts off taller, but Plant A grows at a higher rate. Eventually, Plant A's height will surpass Plant B's height, as illustrated at the end of Week 12 and beyond (since the plants are already the same height at that point).
In conclusion, only the second statement ("Plant A and Plant B will be the same height at the end of Week 12.") is true.