Question

An ornithologist has studied a particular population of starlings and discovered that their population has increased by \(400 \%\) every ten years starting in \(1890 .\) If the initial population in 1890 was 256 birds, how large was the population of starlings in \(1970 ?\) a. 102,400 b. 10,000,000 c. 16,777,216 d. 20,000,000 e. 100,000,000

Short answer

e. 100,000,000

Step-by-step solution

Calculate the number of 10-year periods

Determine how many 10-year periods have passed from 1890 to 1970. Subtract 1890 from 1970 to get the total number of years, then divide by 10. 1970 - 1890 = 80 years 80 / 10 = 8 periods

Determine the growth factor

The population increases by 400% every ten years. A 400% increase means the population becomes 5 times larger every ten years (since 400% increase means the new population is 500% of the original population).Growth factor per period = 5

Calculate the population in 1970

Use the initial population and the growth factor to find the population in 1970. Raise the growth factor to the power of the number of periods and multiply by the initial population.Initial population = 256 birdsGrowth factor per period = 5Number of periods = 8Population in 1970 = 256 * 5^8= 256 * 390625 = 100,000,000

Select the correct option

The calculated population is 100,000,000, so the correct option is: e. 100,000,000

Key concepts

exponential growth

Exponential growth is a key concept often found in population studies. It describes situations where the growth rate of a population is directly proportional to its current size.
In other words, as a population grows, its rate of growth also increases, leading to faster and faster growth over time. The classic formula for exponential growth is:
\[P(t) = P_0 \times (1 + r)^t\]
Where:

• \(P(t)\): The population at time \(t\)
• \(P_0\): The initial population size
• \(r\): The growth rate
• \(t\): Time period

In the given problem, the growth rate is 400% every ten years, meaning the population multiplies by 5 every decade. This results in a highly rapid increase, showcasing the power of exponential growth.

arithmetic calculations

Arithmetic calculations are essential for solving exponential growth problems. The first step is to determine the number of time intervals. For the exercise, the interval is ten years.
Starting from 1890 up to 1970, we calculate:
\[1970 - 1890 = 80 \text{ years}\]
Next, divide by the interval length to find the number of periods:
\[80 / 10 = 8 \text{ periods}\]
Once we know the number of 10-year periods, we use the growth factor. Multiplying the initial population by the growth factor raised to the power of the number of periods gives us the final population:
\[P_{1970} = 256 \times 5^8 = 100,000,000\]
Careful and systematic arithmetic calculations ensure that we arrive at the correct solution.

standardized test preparation

Preparing for standardized tests like the GMAT involves understanding various types of problems, including those involving population growth. Here are some steps to help you tackle such problems effectively:

• Understand the problem: Carefully read all given information.
• Identify the required formula: For population growth, it's often exponential growth formulas.
• Break down the problem: Divide it into manageable steps.
• Perform calculations: Do the math systematically, step-by-step.
• Double-check: Revalidate your answers and calculations.

Following these steps can ensure accuracy and efficiency on the test day, helping you solve even the trickiest of problems.

population modeling

Population modeling uses mathematical techniques to predict population changes over time. In the given problem, we used exponential growth to model the starling population. This method is widely applicable in biology, ecology, and even in finance.
Key elements of population modeling include:

• Initial population size
• Growth rate
• Time intervals
• Mathematical formulas to predict future population sizes

In exponential growth models like ours, the population grows by a constant factor in regular intervals. By using these models, researchers can make informed predictions and decisions.