Question

Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Cell growth The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?

Short answer

Answer: It will take approximately 28.44 weeks for the tumor to reach 1500 cells.

Step-by-step solution

Find the rate constant k

Since the number of cells in the tumor doubles every 6 weeks, we can write: 2 = e^(6k), where 2 is the doubling factor and 6 is the time it takes for the cells to double (6 weeks). To find k, we can take the natural logarithm of both sides of the equation: ln(2) = ln(e^(6k)) ln(2) = 6k. Now, we can solve for k: k = ln(2)/6 ≈ 0.1155.

Derive the exponential growth function

Now that we have the rate constant k, we can write the exponential growth function that describes the number of cells in the tumor, N(t), as a function of time t (in weeks): N(t) = N_0 * e^(kt), where N_0 is the initial number of cells (8) and k is the rate constant we found in step 1. So, the growth function is: N(t) = 8 * e^(0.1155t).

Solve for the time when the number of cells reaches 1500

We want to find the number of weeks it takes for the tumor to have 1500 cells, so we need to solve for t when N(t) = 1500: 1500 = 8 * e^(0.1155t). To solve for t, we can first divide both sides of the equation by 8: 187.5 = e^(0.1155t). Now, we can take the natural logarithm of both sides of the equation: ln(187.5) = ln(e^(0.1155t)) ln(187.5) = 0.1155t. Finally, we can solve for t: t ≈ ln(187.5)/0.1155 ≈ 28.44 weeks. So, after about 28.44 weeks, the tumor will have approximately 1500 cells.

Key concepts

Rate Constant

In exponential growth, the rate constant, denoted as \( k \), plays a crucial role in determining how quickly a quantity grows over time. It's a measure of how fast the growth occurs for a particular process. The rate constant is found using information about how often the quantity in question doubles. In our cell growth example, we know that the cells double every 6 weeks.
To find \( k \), we use the equation \( 2 = e^{6k} \), because it represents the doubling process mathematically. Taking the natural logarithm of both sides helps us solve for \( k \):
\[ \ln(2) = 6k \]
So, \( k = \ln(2)/6 \approx 0.1155 \).
This constant is fundamental in developing any exponential growth model because it defines the specific characteristics of how fast the growth happens.

Doubling Time

Doubling time is the period it takes for a quantity experiencing exponential growth to double in size or value. This concept is crucial for understanding and predicting the growth patterns of populations, investments, and even biological entities, like tumors or bacterial cultures.
When you know the doubling time—as in the exercise, where cells double every 6 weeks—you can predict future values. The doubling time can be directly linked to the rate constant \( k \) with the equation:
\[ 2^{(t/T_d)} = e^{kt} \]
where \( T_d \) is the doubling time. Here, \( T_d = 6 \) weeks corresponds with \( 2 = e^{6k} \), validating the calculation for \( k \). Understanding the doubling time helps you better grasp how rapidly change occurs within exponential growth scenarios.

Cell Growth

Cell growth is a biological process whereby cells multiply and increase in number. When modeled mathematically as exponential growth, we can predict how cell populations change over time. This modeling is essential for situations like estimating time to reach certain cell counts in a tumor.
In this exercise, we begin with an initial number of cells \( N_0 = 8 \). Using exponential growth laws, the cell population follows the formula:
\[ N(t) = N_0 \times e^{kt} \]
where \( k \) is the rate constant. Over time \( t \), this function can predict how many cells will be present. For instance, the growth function \( N(t) = 8 \times e^{0.1155t} \) allows us to solve for any future cell count, providing insights for biological or medical predictions.

Exponential Function

The exponential function \( N(t) = N_0 \times e^{kt} \) is a mathematical model used to describe exponential growth processes. It's characterized by a constant rate of growth, which leads to rapid increases as time progresses.
In general, these functions start with an initial value, \( N_0 \), which is then multiplied by a factor \( e^{kt} \), highlighting the effect of the growth rate \( k \) over time \( t \).
Exponential functions are powerful tools for modeling various real-world phenomena, such as population dynamics, radioactive decay, and financial investments. They provide important insights, especially when the systems involved exhibit consistent proportional growth relative to their size. Understanding these functions helps in both mathematical studies and practical applications where predicting future outcomes is essential.