A colony of ants is growing so that the total number of ants present, A, after a

number of weeks, t, is shown in the table.
t 5 10 15 20 25 30
A 140 200 280 400 570 800
The food supply available is such that it can feed A ants at any given time, t (weeks),according to the table below.
t 5 10 15 20 25 30
A 440 480 520 560 600 640
At what point in time are the ant population and food supply balanced?
Fully justify your response.

Interpolation polynomial for first set :

( x ^ 5 - 100 x ^ 4 + 4375 x ^ 3 - 65000 x ^ 2 + 827500 x + 2250000 ) / 37500

Interpolation polynomial for second set :

8 x + 400

( x ^ 5 - 100 x ^ 4 + 4375 x ^ 3 - 65000 x ^ 2 + 827500 x + 2250000 ) / 37500 = 8 x + 400

Real solution :

x = 25.91833

Can you please explain how you solved this equation? Im confused.

To find the point in time when the ant population and food supply are balanced, we need to compare the values of ants present (A) and the available food supply (F) at each week (t) listed in the table.

We can start by comparing the values at week 5. According to the table:

At week 5:
A = 140 ants
F = 440 units of food supply

Since the ant population is smaller than the available food supply, the two are not balanced.

We can continue this comparison for each week:

At week 10:
A = 200 ants
F = 480 units of food supply

At week 15:
A = 280 ants
F = 520 units of food supply

At week 20:
A = 400 ants
F = 560 units of food supply

At week 25:
A = 570 ants
F = 600 units of food supply

At week 30:
A = 800 ants
F = 640 units of food supply

By comparing the values, we see that at week 30, the number of ants (A = 800) and the available food supply (F = 640) are equal. This is the point in time when the ant population and food supply are balanced.

To fully justify this response, we need to show that before week 30, the ant population was growing faster than the available food supply. This is evident from the increasing values of A compared to F in the earlier weeks. After week 30, the ant population continues to grow but at a slower rate compared to the available food supply.

Therefore, at week 30, the ant population and food supply are balanced.