At a bird sanctuary, the population of bluebirds is 120 and is increasing 4% annually. The population of goldfinches is 400 and is decreasing 2% annually.
a. Write and graph exponential functions to model the populations of the two types of birds as a function of years.
b. In about how many years will the two bird populations be approximately the same? c. What will the approximate populations be at that point in time?
To model the population of bluebirds as a function of years, we can use the exponential growth formula:
P(t) = P(0) * (1 + r)^t,
where P(t) represents the population at time t, P(0) is the initial population, r is the growth rate (expressed as a decimal), and t is the number of years.
In this case, the initial population of bluebirds (P(0)) is 120 and the growth rate (r) is 4% or 0.04. Therefore, the exponential function that models the population of bluebirds is:
P(t) = 120 * (1 + 0.04)^t.
Similarly, to model the population of goldfinches as a function of years, we can use the exponential decay formula:
P(t) = P(0) * (1 - r)^t,
where P(t) represents the population at time t, P(0) is the initial population, r is the decay rate (expressed as a decimal), and t is the number of years.
In this case, the initial population of goldfinches (P(0)) is 400 and the decay rate (r) is 2% or 0.02. Therefore, the exponential function that models the population of goldfinches is:
P(t) = 400 * (1 - 0.02)^t.
To graph these functions, we can plot the population on the y-axis and the number of years on the x-axis.
To find the approximate number of years when the two bird populations will be approximately the same, we need to find the point at which the two exponential functions intersect. We can do this by setting the two functions equal to each other and solving for t:
120 * (1 + 0.04)^t = 400 * (1 - 0.02)^t.
Simplifying, we get:
(1.04)^t = (0.98)^t.
By taking the natural logarithm of both sides, we get:
t * ln(1.04) = t * ln(0.98).
The t's will cancel out, so we're left with:
ln(1.04) = ln(0.98).
Using a calculator, we find that ln(1.04) ≈ 0.0392 and ln(0.98) ≈ -0.0202.
Therefore, t ≈ -0.0202 / 0.0392 ≈ -0.52.
Since we can't have a negative number of years, we can conclude that the two bird populations will be approximately the same in about 0.52 years or approximately 6 months.
To find the approximate populations at that point in time, we can substitute t = 0.52 into either of the exponential functions. Let's use the exponential growth function for bluebirds:
P(0.52) ≈ 120 * (1 + 0.04)^0.52 ≈ 120 * (1.04)^0.52 ≈ 120 * 1.021006 ≈ 122.52.
Therefore, the approximate populations at that point in time are approximately 122 bluebirds and 400 goldfinches.
a. We can model the population of bluebirds using the exponential growth formula:
Bluebird population = Initial population * (1 + growth rate)^time
where the initial population is 120 and the growth rate is 4% (or 0.04) annually.
The exponential function for bluebirds is:
B(t) = 120 * (1 + 0.04)^t
We can model the population of goldfinches using the exponential decay formula:
Goldfinch population = Initial population * (1 - decay rate)^time
where the initial population is 400 and the decay rate is 2% (or 0.02) annually.
The exponential function for goldfinches is:
G(t) = 400 * (1 - 0.02)^t
To graph these functions, let's plot the population on the y-axis and time (in years) on the x-axis.
b. To find the point in time when the two populations are approximately the same, we need to find the time value where B(t) = G(t). Let's solve this equation:
120 * (1 + 0.04)^t = 400 * (1 - 0.02)^t
To estimate the value of t, we can use trial and error or use a graphing calculator.
c. Once we find the value of t, we can substitute it back into either B(t) or G(t) to find the approximate populations at that point in time.