The population of a small town is modeled by the function p(t)=20(4t+3)/2t+5, where P(t) is the population, in thousands, and t is time, in years, since the start of 1990. The town will need its own transit system if the population exceeds 50,000. Will the town's population ever exceed 50,000? Explain.
as t gets very large p(t)---> 20(4/2)
or 40,000
To determine if the town's population will ever exceed 50,000, we need to find the value of t for which p(t) is greater than 50 (since the population is given in thousands).
The population function is given as p(t) = 20(4t + 3) / (2t + 5).
To solve p(t) > 50, we'll start by setting the inequality:
20(4t + 3) / (2t + 5) > 50
Now let's simplify the inequality:
(4t + 3) / (2t + 5) > 2.5
Multiply both sides of the inequality by (2t + 5) to cancel out the denominator:
(4t + 3) > 2.5(2t + 5)
Simplify the right side:
(4t + 3) > 5t + 12.5
Next, subtract 5t from both sides:
4t + 3 - 5t > 12.5
Simplify:
-t + 3 > 12.5
Subtract 3 from both sides:
-t > 9.5
Multiply both sides by -1 and change the direction of the inequality:
t < -9.5
So, the inequality tells us that the population will exceed 50,000 when t is less than -9.5.
However, since t represents time since the start of 1990, a negative value for t does not make sense in this context. Therefore, the solution is invalid, and the town's population will not exceed 50,000.
In other words, based on the given population function, the town will not require its own transit system.