Over time, the number of organisms in a population increases exponentially. The table below shows the approximate number of organisms after y years.
| y years | number of organisms, n |
| ------- | ---------------------- |
| 1 | 55 |
| 2 | 60 |
| 3 | 67 |
| 4 | 75 |
The environment in which the organism lives 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. 61
D. 82
Answer (2)
The population of organisms will theoretically exceed the environment's carrying capacity of 600 organisms in approximately 28.73 years. This indicates it will be well beyond the provided options of 12, 24, 61, or 82 years. The correct rounding indicates that the population would exceed this much earlier than the choices allow and suggests a misalignment with options presented.
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Model the exponential growth: n = a × b y .
Calculate b using points (1, 55) and (2, 60): b = 55 60 = 11 12 .
Calculate a : a = 55 × 12 11 = 12 605 .
Solve for y when n = 600 : y = l n ( 11 12 ) l n ( 605 7200 ) ≈ 28.46 . Therefore, the answer is 28.46 .
Explanation
Understanding the Problem We are given a table showing the number of organisms in a population over time, and we are told that the population increases exponentially. Our goal is to find out after how many years the population will exceed the environment's carrying capacity of 600 organisms.
Finding the Exponential Model First, we need to find an exponential function that models the population growth. The general form of an exponential function is n = a × b y , where n is the number of organisms, y is the number of years, a is the initial population, and b is the growth factor. We can use the data points from the table to estimate the values of a and b .
Calculating the Growth Factor (b) Let's use the first two data points, (1, 55) and (2, 60), to find a and b . We have the following equations:
55 = a × b 1 (1)
60 = a × b 2 (2)
Dividing equation (2) by equation (1), we get:
55 60 = a × b a × b 2
55 60 = b
b = 11 12 ≈ 1.0909
Calculating the Initial Population (a) Now, substitute the value of b back into equation (1) to find a :
55 = a × 11 12
a = 55 × 12 11
a = 12 605 ≈ 50.4167
The Exponential Model So, our exponential function is approximately:
n = 12 605 × ( 11 12 ) y
Setting up the Equation Now we want to find the number of years, y , when the population reaches 600. Set n = 600 and solve for y :
600 = 12 605 × ( 11 12 ) y
605 600 × 12 = ( 11 12 ) y
605 7200 = ( 11 12 ) y
Solving for y Take the natural logarithm of both sides:
ln ( 605 7200 ) = y × ln ( 11 12 )
Solve for y :
y = l n ( 11 12 ) l n ( 605 7200 )
y ≈ l n ( 1.0909 ) l n ( 11.89 )
y ≈ 0.08617 2.476 ≈ 28.73
Final Answer Since y represents the number of years, we round to the nearest whole number. Therefore, it will take approximately 28.73 years for the population to reach 600. Since the question asks after how many years the environment will no longer be able to support the population, we round up to the next whole year, which is 29. However, since 28.73 is closer to 28 than 29, and the options provided are 12, 24, 61, and 82, we should check the value of the function at these points to see which is closest to 600. Since we have already calculated the value to be approximately 28.73, we can conclude that the closest answer is approximately 29.
Final Calculation Based on the calculation using the exponential model derived from the given data, the environment will no longer be able to support the population after approximately 28.46 years.
Examples
Exponential growth models are used in various real-world scenarios, such as predicting population growth, calculating compound interest, and modeling the spread of diseases. For example, if you invest money in a savings account with a fixed interest rate, the amount of money in your account will grow exponentially over time. Similarly, the number of people infected during an epidemic can also grow exponentially in the early stages of the outbreak. Understanding exponential growth helps us make informed decisions in finance, public health, and other fields.
