Problem 60 - College Algebra

Question

A scientist places 50 cells in a petri dish. Every hour, the population increases by \(1.5 \% .\) What will the cell count be after 1 day?

Step-by-Step Solution

Answer

The cell count after 1 day will be 71.

Step 1: Determine the Growth Factor

The population increases by \(1.5\%\) every hour. To find the growth factor, we convert the percentage increase to a decimal fraction and add it to 1. Thus, the growth factor is \(1 + 0.015 = 1.015\).

Step 2: Calculate Total Number of Hours

Since we are asked to find the cell count after one day, we need to calculate the total number of hours in a day. There are 24 hours in a day.

Step 3: Apply the Compound Growth Formula

To find the population after a certain number of time periods with a constant growth rate, we use the formula \(P = P_0 \times (1 + r)^t\), where \(P_0\) is the initial population, \(r\) is the growth rate per period, and \(t\) is the number of periods. In this case, \(P_0 = 50\), \(r = 0.015\), and \(t = 24\).

Step 4: Calculate the Population After 24 Hours

Substitute the values into the formula: \(P = 50 \times (1.015)^{24}\). \(P = 50 \times 1.4232551 = 71.162755\). Since cells cannot be in fractions, we round the result to the nearest whole number.

Step 5: Final Population Count

After rounding \(71.162755\) to the nearest whole number, the population of cells after 1 day is \(71\).

Key Concepts

compound interest formula

The compound interest formula is a powerful mathematical equation used not only in finance but in various fields like biology for growth modeling. It calculates the growth of an initial quantity over time, by applying a consistent growth rate at regular intervals. The formula is given by:\[P = P_0 \times (1 + r)^t\]Where:

• \(P\) represents the future value of the population or investment.
• \(P_0\) is the initial amount or population.
• \(r\) is the growth rate per period (expressed as a decimal).
• \(t\) is the number of periods the growth is applied.

In our exercise, this formula helped determine how a small population of 50 cells could grow over a 24-hour period with a constant hourly growth rate. By substituting the given values into the formula, the cell population after one day was effectively calculated, illustrating the practical application of compound growth dynamics in biological settings.

growth factor

Understanding the growth factor is crucial in models that involve exponential change, like compound interest or population growth. The growth factor is determined by converting the growth rate into a decimal and adding it to one. For a growth rate of \(1.5\%\), the growth factor becomes:\[1.015 = 1 + 0.015\]Here's why this matters:

• The factor \(1.015\) tells us the multiplication factor for the population each interval (in this case, hourly).
• Multiplying the organisms by this factor each hour reflects compounding growth, where growth accumulates on previous growth.
• This concept underscores why small increases in growth rates lead to notable increases over time.

Thus, recognizing and using the growth factor is central to understanding how populations can rapidly expand under continuous, consistent growth conditions.

population modeling

Population modeling involves using mathematical expressions to predict how a group of organisms changes over time. It is essential for fields like ecology, medicine, and conservation. By understanding population dynamics, we can make informed predictions and decisions.In our exercise, population modeling uses the compound growth formula to project the number of cells in a petri dish after a given duration.Key elements in effective population modeling include:

• Initial Population: The starting point, here 50 cells, serves as the baseline for modeling.
• Growth Rate: Determines how quickly the population increases. In this scenario, it’s hourly at \(1.5\%\).
• Time Frame: Defines how long the growth is tracked. We calculated for a 24-hour period resulting in an increased final cell count.

Through this simple yet powerful approach, we can model not just cell growth but many natural phenomena, assisting in resources management and scientific discovery.