Question

There are 975 birds in an aviary. Each month, the number of birds decreases by 7%. There are 350 trees in the aviary. Each month, 7 trees are removed.Part A: Write functions to represent the number of birds and the number of trees in the aviary throughout the months. (4 points)
Part B: How many birds are in the aviary after 12 months? How many trees are in the aviary after the same number of months? (2 points)

Part C: After approximately how many months is the number of birds and the number of trees the same? Justify your answer mathematically. (4 points)

Answer 1

We can let t represent the number of months that have passed. Let's observe first what happens with number of birds and trees for the first 3 months to better understand the function that we'll create.

For the birds in the aviary, since the number decreases by 7%, for the first month it will have  93% of the original number. For the second month, the number of birds left will be 93% of what was left from the first month as shown below.

1st month | B(1) = 0.93(975) 
2nd month| B(2) = 0.93[0.93(975)] = 0.93²(975)
3rd month| B(3) = 0.93[0.93²(975)] = 0.93³(975)

From this, we can see where the pattern is going and conclude that after t months, there will be B(t) = 0.93ᵗ(975) birds present.

We'll use the same approach to estimate the number of trees after t months in the aviary.

T(1) = 350 - 7
T(2) = 350 - 7 - 7 = 350 - 2(7)
T(3) = 350 - 2(7) - 7 = 350 - 3(7)
and hence, 
T(t) = 350 - 7t
Thus, we have the functions for the birds and trees throughout t months. 

We can now compute for the number of birds and trees present after 12 months by simply substituting 12 into our equations as shown below.

B(12) = 0.93¹²(975) = 408.13 ≈ 408
T(12) = 350 - 12(7) =  266

Hence, from this we can see that there are approximately 408 birds and 266 trees present in the aviary after 12 months.

To find the number months when there are almost the same number of trees and birds present, we can equate T(t) to B(t) and solve for the value of t. Using the graphical method, we can also find the intersection points between the two functions.

350 - 7t = 0.93ᵗ(975)

We have intersection points at around 22.21 and 44.4781. Approximating these values to whole numbers, we have 22 and 44 months.

Answers:

Part A: B(t) = 0.93ᵗ(975) and T(t) = 350 - 7t
Part B: 408 birds and 266 trees 
Part C: 22 and 44 months