The population of a small town is modelled 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.
a) Is the population currently increasing or decreasing? Justify your answer.
b) The town will need its own transit system if the population exceeds
50 000. Will the town’s population ever exceed 50 000? Explain.
See previous post: 7:09pm.
To determine if the population is increasing or decreasing, we need to analyze the function p(t). Let's break it down step by step:
a) Is the population currently increasing or decreasing?
First, we need to find the derivative of p(t) with respect to t to determine the rate of change of population over time.
p(t) = 20(4t+3) / (2t+5)
To simplify the derivative calculation, we can rewrite the function as:
p(t) = 20(4t+3) * (2t+5)^(-1)
Now, we differentiate p(t) using the product rule and the chain rule:
p'(t) = 20(4) * (2t+5)^(-1) + 20(4t+3) * (-1)(2t+5)^(-2) * 2
p'(t) = 80 / (2t+5) - 40(4t+3) / (2t+5)^2
To determine if the population is increasing or decreasing, we need to evaluate the sign of p'(t) for a specific value of t. Since we don't have a specific value for t, we can analyze the sign of the derivative expression.
For any given value of t, the denominator (2t+5) is always positive. Therefore, to determine the sign of p'(t), we only need to consider the numerator.
The numerator 80 - 40(4t+3) simplifies to 80 - 160t - 120, which further simplifies to -160t - 40.
Since the coefficient of t, -160, is negative, the sign of p'(t) will be determined by the value of t. If t is positive, -160t will be negative, and vice versa.
Therefore, the population is currently increasing if t < 0 and decreasing if t > 0.
b) Will the town’s population ever exceed 50,000?
To determine if the population will ever exceed 50,000, we need to analyze the function p(t) itself, rather than its derivative.
p(t) = 20(4t+3) / (2t+5)
To determine if the population will exceed 50,000, we need to find the value of t when p(t) is equal to 50.
Set p(t) = 50:
50 = 20(4t+3) / (2t+5)
Now, cross-multiply and solve for t:
50(2t + 5) = 20(4t + 3)
100t + 250 = 80t + 60
20t = -190
t = -9.5
Since t is negative and represents time since the start of 1990, t = -9.5 means 9.5 years before the start of 1990. Therefore, the population will never exceed 50,000 based on this model.
To determine if the population is currently increasing or decreasing, we need to analyze the function p(t)= 20(4t+3)/(2t+5).
a) To determine if the population is increasing or decreasing, we need to find the derivative of the function with respect to time (t). The derivative will indicate the rate of change of the population with respect to time.
Let's find the derivative of p(t) with respect to t:
p'(t) = (d/dt)[20(4t+3)/(2t+5)]
Using the quotient rule, the derivative is calculated as follows:
p'(t) = [(20)((2t+5)(4))-20(4t+3)(2)] / [(2t+5)²]
Simplifying the above expression:
p'(t) = (160t + 400 - 160t - 120) / (4t² + 20t + 25)
p'(t) = 280 / (4t² + 20t + 25)
Since the derivative, p'(t), is positive, it means the population is currently increasing.
b) To determine if the town's population will ever exceed 50,000, we need to set up the inequality p(t) > 50.
20(4t+3)/(2t+5) > 50
Multiplying both sides of the inequality by (2t+5):
20(4t+3) > 50(2t+5)
Simplifying:
80t + 60 > 100t + 250
-20t > 190
t < -9.5
Since the value of t is negative, it means the time is before the start of 1990. Therefore, the population will not exceed 50,000.