The number of Sasquatch living t years after 2016 is given by the formula s(t)=(51t+84)/(3t+28).

-What value does the Sasquatch population approach in the long run? When will the population (rounded to the nearest Sasquatch) reach this value?
(This vale would never be an output value of s(t), but the outputs would be rounded to the nearest whole number. For example, if the limiting value were 53, you would determine when the function's output would first reach at least 52.5).

as t gets huge the 84 and the 28 are small potatoes

s ---> 51/3 = 17

16.5 = (51 t + 84)/(3t+28)
solve for t

To find the value that the Sasquatch population approaches in the long run, we need to find the limit of the function s(t) as t approaches infinity.

In the given function s(t) = (51t + 84) / (3t + 28), as t approaches infinity, the lower powers of t become insignificant compared to the higher powers. So we can neglect the 84 and 28 terms and simplify the equation to:

s(t) = (51t) / (3t) = 17

So, the Sasquatch population approaches 17 in the long run.

Now, let's determine when the population (rounded to the nearest Sasquatch) will reach this value.

To find the time when the population reaches 17, we need to solve the equation:

s(t) = 17
(51t + 84) / (3t + 28) = 17

To solve this equation, we can cross multiply:
(51t + 84) = 17(3t + 28)

Expanding the right side:
51t + 84 = 51t + 476

Rearranging the equation:
51t - 51t = 476 - 84
0 = 392

This equation does not have a valid solution because zero is not equal to 392. Therefore, the population will never reach exactly 17.

However, if we interpret "when the function's output would first reach at least 16.5 (rounded to the nearest whole number)" as the question suggests, we need to find when s(t) becomes greater than or equal to 16.5.

s(t) = (51t + 84) / (3t + 28) ≥ 16.5

Cross multiplying:
(51t + 84) ≥ 16.5(3t + 28)

Expanding the right side:
51t + 84 ≥ 49.5t + 462

Rearranging the equation:
51t - 49.5t ≥ 462 - 84
1.5t ≥ 378

Dividing both sides by 1.5:
t ≥ 252

Therefore, the population will reach or exceed 16.5 Sasquatches approximately 252 years after 2016.