Problem 55 - Calculus Volume 1

Question

A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by \(r(t)=6-\left[5 /\left(t^{2}+1\right)\right]\) where \(t\) is time measured in hours since a circle of a \(1-\mathrm{cm}\) radius of the bacterium was put into the culture. a. Express the area of the bacteria as a function of time. b. Find the exact and approximate area of the bacterial culture in 3 hours. c. Express the circumference of the bacteria as a function of time. d. Find the exact and approximate circumference of the bacteria in 3 hours.

Step-by-Step Solution

Answer

Area at 3 hours is approximately 94.986 cm²; circumference is approximately 34.558 cm.

Step 1: Express the Area as a Function of Time

The area of a circle is given by the formula \( A = \pi r^2 \). Substitute \( r(t) = 6 - \frac{5}{t^2 + 1} \) into this formula. The area as a function of time is \( A(t) = \pi \left( 6 - \frac{5}{t^2 + 1} \right)^2 \).

Step 2: Calculate the Exact Area at 3 Hours

We need to find \( A(3) = \pi \left( 6 - \frac{5}{3^2 + 1} \right)^2 \). Firstly, calculate \( r(3) = 6 - \frac{5}{10} = 6 - 0.5 = 5.5 \). Thus, \( A(3) = \pi (5.5)^2 = \pi \times 30.25 \).

Step 3: Approximate the Area at 3 Hours

Use the approximation \( \pi \approx 3.14159 \). Therefore, \( A(3) \approx 3.14159 \times 30.25 \approx 94.9859 \) square centimeters.

Step 4: Express the Circumference as a Function of Time

The circumference of a circle is given by \( C = 2\pi r \). Substitute \( r(t) = 6 - \frac{5}{t^2 + 1} \) into this formula. The circumference as a function of time is \( C(t) = 2\pi \left( 6 - \frac{5}{t^2 + 1} \right) \).

Step 5: Calculate the Exact Circumference at 3 Hours

Find \( C(3) = 2\pi \times 5.5 \). So, \( C(3) = 11\pi \).

Step 6: Approximate the Circumference at 3 Hours

Use the approximation \( \pi \approx 3.14159 \). Therefore, \( C(3) \approx 11 \times 3.14159 \approx 34.55749 \) centimeters.

Key Concepts

Circular Region

Bacteria often grow in a circular region when placed in cultures. This shape allows them to exploit available space efficiently while limiting any boundary-related waste. In our exercise, the bacterium occupies a growing circular region, governed by the model of its radius over time. The circle’s boundaries imply that measurements such as area and circumference are pertinent for assessing how the bacterium population expands in space. Understanding the dynamics within circular regions can aid in predicting how quickly dangerous or beneficial bacteria spread in lab experiments.

The circular shape emerges naturally from uniform growth patterns, where expansion occurs equally in all directions. Thus, the concepts of area and circumference uniquely represent the spatial extent of bacterial growth from its initial state.

Radius Function

The radius function of the bacterial growth describes how the size of the circle changes over time. In the given scenario, the radius of the circle, denoted as \( r(t) \), is described mathematically by \( r(t)=6-\left[\frac{5}{t^2+1}\right] \). Here, \( t \) represents time in hours since the bacteria were introduced into the culture with an initial radius of 1 cm.

• The term \( 6 \) can be understood as the maximum radius, which the bacteria aim to reach over time.
• The second part, \( \frac{5}{t^2+1} \), allows the function to denote changes as \( t \) increases, dictating a narrowing effect on the radius progression.

This equation tells us that at the start, where \( t = 0 \), the radius grows slowly due to the initial value manipulation by the fraction. As time progresses to infinity, the impact of this fractional term lessens, allowing the radius to approach its theoretical maximum of 6 cm.

Area Function

The area of a circle is linked directly to its radius via the formula \( A = \pi r^2 \). In our problem, after determining the radius function, we substitute \( r(t) = 6 - \frac{5}{t^2 + 1} \) into the area formula. Hence, the area as a function of time becomes \( A(t) = \pi \left(6 - \frac{5}{t^2 + 1}\right)^2 \).

• This function gives insight into how the physical space—occupied by bacteria—evolves as time progresses.
• The square of the radius simplifies the task of identifying the bacterial culture's spatial coverage at any moment.

As seen in the calculations, the exact area at \( t = 3 \) hours is determined by substituting \( t = 3 \) into our area function, delivering \( A(3) = \pi \times (5.5)^2 = \pi \times 30.25 \). By approximating \( \pi \) as 3.14159, we can estimate the area around 94.9859 square centimeters.

Circumference Function

To find the circumference of a circle, the equation \( C = 2 \pi r \) is employed, where \( r \) is the radius. By substituting the radius function \( r(t) = 6 - \frac{5}{t^2 + 1} \) into this formula, we derive the bacterial growth's circumference as \( C(t) = 2 \pi \left(6 - \frac{5}{t^2 + 1}\right) \).

This circumference function allows us to easily assess how the "boundary" of the bacterium's growth region lengthens through time:

• This perimeter value transitions more smoothly across time than the area functions, reinforcing the growth's isotropic pace.
• At \( t = 3 \), calculations show \( C(3) = 11\pi \), which approximates to 34.55749 cm upon simplifying with \( \pi \approx 3.14159 \).

Thus, the circumference function aids in visualizing how the growth edge stretches and adapts as the bacterial culture enlarges.