The function f(t) = 24,500(0.84) models the population of a town t years
after 2008. What rate is the population of the town decreasing each year?
A. 16%
B. 24.5%
C. 75.5%
D. 84%
If f(t) = 24,500(0.84) mean f(t) =24,500 ∙ 0.84ᵗ
then:
t = present year
t - 1 = last year
24,500 ∙ 0.84ᵗ = population in present year
24,500 ∙ 0.84ᵗ⁻¹ = population in last year
Rate of decreasing = 1 - ( 24,500 ∙ 0.84ᵗ / 24,500 ∙ 0.84ᵗ⁻¹ ) = 1 - 0.84ᵗ / 0.84ᵗ⁻¹ = 1 - 0.84 = 0.16 = 16%
To find the rate at which the population of the town is decreasing each year, we need to determine the rate of change of the function f(t).
The given function f(t) = 24,500 * (0.84)^t models the population of the town t years after 2008.
We can express the rate of change as the derivative of f(t) with respect to t.
f'(t) = d/dt [24,500 * (0.84)^t]
To calculate this derivative, we use the chain rule. The derivative of a constant times a function raised to a power is given by:
f'(t) = constant * [ (e^(ln(0.84))) * (d/dt (0.84)^t)]
Applying the chain rule, we obtain:
f'(t) = constant * (0.84^t) * ln(0.84)
Since the problem asks for the rate of decrease, which is a negative value, we need to consider the negative derivative. Therefore, the rate at which the population of the town is decreasing each year is given by:
Rate of decrease = - constant * (0.84^t) * ln(0.84)
Since the constant is not provided, we cannot calculate the exact rate of decrease. However, we can conclude that the correct answer is not one of the options given (A, B, C, or D), as those options represent percentages rather than a mathematical expression.
To find the rate at which the population of the town is decreasing each year, we need to determine the derivative of the given function, f(t).
The given function f(t) = 24,500(0.84) represents the population of the town at any given year t. Since the function is in the form of an exponential decay model (with a base less than 1), the derivative of f(t) will give us the rate of decrease.
To find the derivative, we need to use the rules of differentiation. The derivative of a constant multiplied by a function can be found by multiplying the derivative of the function by the constant.
In this case, f(t) = 24,500(0.84), where f(t) represents the population of the town at year t, and 0.84 represents the base of the exponential decay.
Differentiating the function, we get:
f'(t) = 24,500 * ln(0.84) * (0.84)^t
Now that we have the derivative, we can calculate the rate of decrease at any given year by substituting the value of t in the derivative equation.
To find the rate of decrease each year, we substitute t = 1 into the derivative equation:
f'(1) = 24,500 * ln(0.84) * (0.84)^1
f'(1) ≈ -0.16 * 24,500
f'(1) ≈ -3,920
The negative sign indicates a decrease, and the magnitude of -3,920 represents the rate of decrease in population each year.
To determine the percentage rate of decrease, we can divide the rate of decrease by the initial population and multiply by 100:
percentage rate of decrease = (rate of decrease / initial population) * 100
= (-3,920 / 24,500) * 100
≈ -16%
Therefore, the population of the town is decreasing at a rate of approximately 16% each year.
Therefore, the correct answer is A. 16%.