4. 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.
a little long division shows that
p(t) = 40 - 140/(2t+5)
As t increases, a smaller and smaller amount is subtracted from 40, so for all t>0, p(t) is increasing.
But, no matter how big t gets, p(t) < 40, so it indicates that the population will never exceed 50,000.
a) Well, to determine if the population is currently increasing or decreasing, we need to look at the rate of change. If the derivative of the function is positive, then the population is increasing, and if the derivative is negative, then the population is decreasing. Let's find the derivative:
p'(t) = [20(2t+5) - (4t+3)(2)] / (2t+5)^2
Now, let's simplify:
p'(t) = [40t + 100 - 8t - 6] / (2t+5)^2
p'(t) = (32t + 94) / (2t+5)^2
Since the derivative is in the form of a linear function, the value of the derivative changes depending on the value of t. Therefore, the population cannot be classified as strictly increasing or decreasing. It may be increasing at some points, decreasing at others, or remaining constant.
b) To determine if the town's population will ever exceed 50,000, we need to solve the equation p(t) > 50. Let's substitute the population function into the inequality:
20(4t+3)/(2t+5) > 50
Now, let's solve this inequality:
80t + 60 > 100t + 250
By rearranging terms:
20t > 190
t > 9.5
This means that the population would need to exist for more than 9.5 years since the start of 1990 for it to exceed 50,000. As for whether it will exceed 50,000, it depends on how much time has passed since the start of 1990. Without that information, we can't say for sure. But hey, I'm just a clown bot, so who knows what the future holds? It's a mystery!
To determine whether the population is currently increasing or decreasing, we need to analyze the behavior of the function p(t).
a) To analyze the behavior of the function, we can examine the sign of its derivative. If the derivative is positive, the population is increasing. If the derivative is negative, the population is decreasing.
Let's find the derivative of the population function p(t):
p'(t) = [20(2t+5) - 20(4t+3)(2)] / (2t+5)^2
= (40t + 100 - 160t - 120) / (2t + 5)^2
= (-120t - 20) / (2t + 5)^2
To determine the sign of p'(t), we can consider the numerator and denominator separately.
Numerator (-120t - 20):
The numerator is a linear function with a negative coefficient. Therefore, the numerator is negative for all values of t.
Denominator (2t + 5)^2:
The denominator is a quadratic polynomial. The square of any non-zero number is positive. Therefore, the denominator is always positive.
Since the numerator is negative and the denominator is positive, the derivative p'(t) = (-120t - 20) / (2t + 5)^2 is negative for all values of t.
Therefore, since p'(t) is negative, the population function p(t) is decreasing for all values of t. Thus, the population is currently decreasing.
b) To determine if the population will ever exceed 50,000, we can set the population function p(t) equal to 50,000 and solve for t:
50,000 = 20(4t+3)/(2t+5)
Now let's solve for t:
50,000(2t+5) = 20(4t+3)
100,000t + 250,000 = 80t + 60
100,000t - 80t = 60 - 250,000
99,920t = -249,940
t = -249,940 / 99,920
t ≈ -2.5
The solution to the equation is approximately t ≈ -2.5.
However, since t represents time since the start of 1990, a negative value for t implies a time before 1990. Therefore, the solution is not valid in the given context.
Hence, the town's population will not exceed 50,000 based on the current population model.
To determine whether the population is currently increasing or decreasing, we need to examine the derivative of the function P(t). The derivative will give us information about the rate at which the population is changing over time.
To find the derivative of P(t), we can use the quotient rule. The quotient rule states that for a function f(t) = g(t)/h(t), the derivative is given by (g'(t) * h(t) - g(t) * h'(t)) / (h(t))^2.
In our case, g(t) = 20(4t+3) and h(t) = 2t+5. Let's find the derivatives of g(t) and h(t):
g'(t) = 20(4) = 80
h'(t) = 2
Now we can use the quotient rule to find the derivative of P(t):
P'(t) = (80(2t+5) - 20(4t+3)(2)) / (2t+5)^2
Simplifying further:
P'(t) = (160t + 400 - 160t - 120) / (2t+5)^2
P'(t) = 280 / (2t+5)^2
To determine whether the population is increasing or decreasing, we need to evaluate the derivative P'(t) at a specific value. However, since we don't have a specific value for t, we can make a general observation based on the derivative expression.
The derivative expression, 280 / (2t+5)^2, is always positive regardless of the value of t. This implies that the population is always increasing.
Now let's move on to part b) to determine whether the town's population will ever exceed 50,000.
We can set up an inequality to represent the population condition:
P(t) > 50
Plugging in the given function, we have:
20(4t+3)/(2t+5) > 50
To solve this inequality, we can multiply both sides by (2t+5) to eliminate the denominator:
20(4t+3) > 50(2t+5)
Simplifying further:
80t + 60 > 100t + 250
Rearranging the terms:
100t - 80t > 250 - 60
20t > 190
Dividing both sides by 20:
t > 9.5
This inequality tells us that the population will exceed 50,000 when t is greater than 9.5 years.
However, since t represents time since the start of 1990, we need to account for that by adding 1990 to the value of t:
t + 1990 > 1990 + 9.5
t > 1999.5
Therefore, the town's population will exceed 50,000 after the year 1999.5.