Problem 37
Question
For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
Step-by-Step Solution
Verified Answer
The population after 3 hours is 691200 bacteria.
1Step 1: Understanding the Situation
We are given two separate scenarios. The first one is about the decay of Iodine-125, but the task revolves around a second scenario involving bacterial growth. The bacteria double in size every 20 minutes, and the initial population is 1350 bacteria.
2Step 2: Determine the Growth Rate Formula
For exponential growth, the formula is \( P(t) = P_0 \times (1 + r)^t \), where \( P(t) \) is the population after time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate per time period, and \( t \) is the number of time periods.
3Step 3: Calculate the Growth Factor
The bacteria double every 20 minutes, so the growth factor per 20 minutes is 2. Therefore, \( r = 2 - 1 = 1 \) (which is essentially just doubling). The population doubles every 20 minutes, equivalent to a growth rate \( r = 1 \) for the 20-minute interval.
4Step 4: Set Up the Exponential Function
First, find out how many 20-minute periods there are in 3 hours. Since there are 60 minutes in an hour, 3 hours corresponds to \( 3 \times 60 = 180 \) minutes. Dividing 180 by 20 gives 9 periods. The exponential equation for this situation is \( P(t) = 1350 \times 2^t \), where \( t \) is measured in 20-minute periods.
5Step 5: Calculate the Population After 3 Hours
Plug \( t = 9 \) (because there are 9 periods in 3 hours) into the exponential function: \[ P(9) = 1350 \times 2^9 \]. Calculate \( 2^9 = 512 \), then multiply by 1350: \( 1350 \times 512 = 691200 \).
6Step 6: Round the Result
The problem asks for rounding to the nearest whole number, so our calculated population size is already an integer: 691200.
Key Concepts
Bacterial GrowthGrowth Rate CalculationExponential Equation
Bacterial Growth
Bacterial growth describes the increase in a bacterial population over time. This growth typically happens in an exponential manner because bacteria reproduce by binary fission. This means that one bacterium splits into two, two split into four, and so on. As a result, the rate of increase in number is proportional to the current size of the population.
In scenarios like the one in our exercise, the key point is that bacterial populations can double in a set amount of time. Here, the bacteria double every 20 minutes, which is a typical feature of exponential growth. This rapid doubling leads to a dramatic increase in population size over time, especially when considered over several hours.
In scenarios like the one in our exercise, the key point is that bacterial populations can double in a set amount of time. Here, the bacteria double every 20 minutes, which is a typical feature of exponential growth. This rapid doubling leads to a dramatic increase in population size over time, especially when considered over several hours.
Growth Rate Calculation
To understand growth rate calculation in the context of exponential growth, we rely on the formula for exponential growth, which is given by:
- \( P(t) = P_0 \times (1 + r)^t \)
- \( P(t) \) is the population size after time \( t \).
- \( P_0 \) is the initial population size.
- \( r \) is the growth rate per time interval.
- \( t \) is the number of time intervals.
Exponential Equation
The exponential equation is pivotal in modeling scenarios where growth is not constant but rather increases rapidly over time. An exponential equation characterizes a situation where the rate of change is proportional to the current value, such as the doubling of bacterial populations.
For the exercise, we derived the exponential equation for bacterial growth as \( P(t) = 1350 \times 2^t \). Here, 1350 is the initial count of bacteria, and \( 2^t \) represents the doubling effect with each 20-minute period. In this form of the equation, \( t \) is the number of 20-minute growth intervals. For instance, after 9 intervals (or 3 hours), you can calculate the population size as\[ P(9) = 1350 \times 2^9 = 691200 \].
This equation gives us a powerful tool to predict future population sizes based solely on their exponential nature.
For the exercise, we derived the exponential equation for bacterial growth as \( P(t) = 1350 \times 2^t \). Here, 1350 is the initial count of bacteria, and \( 2^t \) represents the doubling effect with each 20-minute period. In this form of the equation, \( t \) is the number of 20-minute growth intervals. For instance, after 9 intervals (or 3 hours), you can calculate the population size as\[ P(9) = 1350 \times 2^9 = 691200 \].
This equation gives us a powerful tool to predict future population sizes based solely on their exponential nature.
Other exercises in this chapter
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