Problem 73 - Applied Calculus

Question

BIOMEDICAL: Cell Growth The number of cells in a culture after \(t\) days is given by \(N(t)=200+50 t^{2}\). Find the size of the culture after: a. 2 days. b. 10 days.

Step-by-Step Solution

Answer

400 cells after 2 days; 5200 cells after 10 days.

Step 1: Understanding the Function

The function that describes the number of cells, \(N(t)\), in the culture after \(t\) days is given by \(N(t) = 200 + 50t^2\). Our task is to find \(N(t)\) for different values of \(t\).

Step 2: Calculating for 2 Days

To find the size of the culture after 2 days, substitute \(t = 2\) into the function \(N(t) = 200 + 50t^2\). This gives us:\[ N(2) = 200 + 50(2^2) = 200 + 50 imes 4 = 200 + 200 = 400. \]

Step 3: Calculating for 10 Days

Next, substitute \(t = 10\) into the function to find the size of the culture after 10 days: \(N(10) = 200 + 50(10^2)\). Calculate the value as follows:\[ N(10) = 200 + 50 imes 100 = 200 + 5000 = 5200. \]

Key Concepts

Cell Growth

Cell growth in biological contexts often refers to the process by which cells increase in size or number. In many scientific studies, understanding how a population of cells grows over time is crucial. This can help in researching how diseases progress or how effective a treatment might be.

The function used to represent cell growth in this problem is quadratic, specifically noted as:

• \(N(t) = 200 + 50t^2\)

Here, \(N(t)\) indicates the number of cells after \(t\) days. The initial number of cells is 200. The term \(50t^2\) indicates how the growth rate accelerates as time increases. This model assumes that the number of cells will not decrease, simulating an ideal growth scenario.

Understanding such growth patterns is essential in applications such as cancer research, where abnormal cell growth needs careful monitoring.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest degree of the variable is two. The general form of a quadratic function is

• \(f(x) = ax^2 + bx + c\)

where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. These functions graph as parabolas, which can open upwards or downwards.

In the cell growth problem, the formula \(N(t) = 200 + 50t^2\) is a quadratic function. Here, the parabola opens upwards because the coefficient of \(t^2\) (which is 50) is positive. This characteristic of quadratic functions is vital in understanding how cell growth accelerates over time.

Quadratic equations are frequently used in various fields such as physics, engineering, and economics to model situations where growth or decay rates are not constant.

Function Evaluation

Function evaluation involves finding the output value of a function for a particular input. It is a straightforward but fundamental concept in mathematics and applied calculus.

To evaluate a function, you substitute the input value into the function and simplify. Let's see two evaluations we performed:

• For 2 days: Substitute \(t = 2\) in \(N(t) = 200 + 50t^2\), so \(N(2) = 200 + 50 \times 2^2 = 400\).
• For 10 days: Substitute \(t = 10\), thus \(N(10) = 200 + 50 \times 10^2 = 5200\).

Function evaluation is incredibly useful as it allows us to compute results for scenarios described by a specific formula. In applied calculus and many real-world applications, being able to quickly evaluate functions is a necessity. By mastering this skill, students can solve a wide range of problems efficiently.