Over time, the number of organisms in a population increases exponentially. The table below shows the approximate number of organisms after [tex]$y$[/tex] years.
| y years | number of organisms, [tex]$n$[/tex] |
| :------ | :----------------------------- |
| 1 | 55 |
| 2 | 60 |
| 3 | 67 |
| 4 | 75 |
The environment in which the organisms live can support at most 600 organisms. Assuming the trend continues, after how many years will the environment no longer be able to support the population?
A. 12
B. 24
C. 81
D. 82
Answer (1)
Set up an exponential function n = a × b y to model the population growth.
Use the given data points (1, 55) and (2, 60) to find the values of a and b , where a ≈ 50.4167 and b ≈ 1.0909 .
Set up the inequality 600"> 12 605 × ( 11 12 ) y > 600 to find when the population exceeds 600.
Solve for y and round up to the nearest whole number, resulting in y = 29 .
$\boxed{29}
Explanation
Understanding the Problem We are given a table showing the approximate number of organisms in a population after y years. The population increases exponentially. The environment can support at most 600 organisms. We need to find the number of years after which the environment can no longer support the population.
Finding the Exponential Function Let's analyze the data to find an exponential function that fits the data. We can assume the exponential function is of the form n = a × b y , where n is the number of organisms after y years, a is the initial number of organisms, and b is the growth factor.
Calculating the Growth Factor Using the data points (1, 55) and (2, 60), we can set up two equations:
55 = a × b 1
60 = a × b 2
Dividing the second equation by the first, we get:
55 60 = a × b 1 a × b 2
11 12 = b
So, b = 11 12 ≈ 1.0909 .
Calculating the Initial Number of Organisms Now, we can find a using the first equation:
55 = a × 11 12
a = 12 55 × 11 = 12 605 ≈ 50.4167
The Exponential Function So, the exponential function is approximately:
n = 12 605 × ( 11 12 ) y
Setting up the Inequality We want to find the number of years y when the population exceeds 600 organisms. So, we set up the inequality:
600"> 12 605 × ( 11 12 ) y > 600
Simplifying the Inequality Divide both sides by 12 605 :
\frac{600 \times 12}{605} = \frac{7200}{605} \approx 11.9008"> ( 11 12 ) y > 605 600 × 12 = 605 7200 ≈ 11.9008
Taking the Logarithm Take the natural logarithm of both sides:
\ln\left(\frac{7200}{605}\right)"> y × ln ( 11 12 ) > ln ( 605 7200 )
Solving for y Solve for y :
\frac{\ln\left(\frac{7200}{605}\right)}{\ln\left(\frac{12}{11}\right)}"> y > l n ( 11 12 ) l n ( 605 7200 )
\frac{\ln(11.9008)}{\ln(1.0909)}"> y > l n ( 1.0909 ) l n ( 11.9008 )
\frac{2.4764}{0.0870}"> y > 0.0870 2.4764
Calculating y 28.46"> y > 28.46
Final Answer Since y must be an integer, we round up to the nearest whole number, which is 29. Therefore, after 29 years, the environment will no longer be able to support the population.
Examples
Exponential growth is a mathematical transformation that grows without bound. An example of this is in the spread of viral diseases, such as the flu. If one person has the flu and they spread it to two people, and those two people spread it to four people, and so on, the number of people with the flu grows exponentially. Understanding exponential growth can help us predict how quickly a disease will spread and take steps to prevent it.
