AP Precalculus Practice Exam Question 77

NumWorks
23 May 202304:07
EducationalLearning
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TLDRThe video script discusses a simulation modeling the growth of a bacterial colony using a geometric sequence. On day four, the population was 4,000 bacteria, and by day eight, it had grown to 49,000. To find the population on day six, the script calculates the common ratio 'r' using the fourth and eighth day populations, resulting in r being the fourth root of 49/4. Applying this ratio to the initial population, the script predicts 14,000 bacteria on day six, engaging viewers in the mathematical modeling of biological growth.

Takeaways
  • 🦠 The population of a bacterial colony can be modeled using a geometric sequence.
  • 📅 The first day of the simulation is considered day one.
  • 🔢 On day four, the population was 4,000 bacteria.
  • 📈 On day eight, the population increased to 49,000 bacteria.
  • ❓ The task is to find the population on day six.
  • 🔄 A geometric sequence model is used: P(t) = P(0) * R^t, where t is time in days.
  • 🤔 The initial population (P(0)) is unknown, but day four (P(4)) is 4,000.
  • ➗ To find the growth rate (R), divide the population on day eight by the population on day four: 49,000 / 4,000.
  • √ Take the fourth root of the result to find R, which is √14/2.
  • 🧮 Use the growth rate to calculate the population on day six: 4,000 * (√14/2)^2, resulting in 14,000 bacteria.
Q & A

  • What type of sequence is used to model the population of the bacterial colony in the simulation?

    -The population of the bacterial colony is modeled using a geometric sequence.

  • What was the population of the colony on day four according to the simulation?

    -The population on day four was 4,000 bacteria.

  • What was the population of the colony on day eight according to the simulation?

    -The population on day eight was 49,000 bacteria.

  • What is the general formula for a geometric sequence in terms of population and time?

    -The general formula for a geometric sequence in terms of population and time is P(T) = P0 * R^T, where P0 is the initial population, R is the rate of change, and T is the time in days.

  • How was the initial population (P0) determined in the simulation?

    -The initial population (P0) was determined by using the population on day four as P0 and setting T to 4.

  • What mathematical operation was used to find the rate of change (R) in the geometric sequence?

    -The fourth root was used to find the rate of change (R) after dividing the population on day eight by the population on day four.

  • What is the value of R after calculating it from the given data?

    -The value of R is √14/2 after taking the fourth root of 49/4.

  • What day's population was the simulation trying to determine after finding R?

    -The simulation was trying to determine the population of the colony on day six.

  • How was the population on day six calculated using the geometric sequence formula?

    -The population on day six was calculated by taking the initial population (4,000), multiplying it by the rate of change (√14/2), and raising it to the power of 2, which represents the two days after day four.

  • What was the final answer for the population of the colony on day six based on the simulation?

    -The final answer for the population of the colony on day six was 14,000 bacteria.

  • What does the result of the simulation suggest about the growth rate of the bacterial colony?

    -The result suggests that the bacterial colony is growing at a rapid rate, as indicated by the large increase in population from day four to day eight.

Outlines
00:00
📊 Geometric Sequence in Bacterial Population Modeling

This paragraph discusses a simulation where the growth of a bacterial colony is represented by a geometric sequence. The initial population is unknown, but the population on the fourth day is given as 4,000 bacteria, and on the eighth day, it is 49,000. The objective is to determine the population on the sixth day. The speaker uses the known data points to establish the growth rate (R) by dividing the population on the eighth day by that on the fourth, resulting in R being the fourth root of 49. The speaker then applies this rate to calculate the population on the sixth day by using the formula P = P₀ * R^T, where T is the number of days after the initial day. By substituting the values and solving, the speaker finds that the population on the sixth day is 14,000 bacteria.

Mindmap
Keywords
💡geometric sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. In the video, the population of the bacterial colony is modeled using a geometric sequence, which is crucial for understanding how the population grows over time. The script uses the terms 'day four' and 'day eight' to illustrate the progression of the sequence, showing how the population increases from 4,000 to 49,000 bacteria.
💡initial population
The initial population refers to the starting number of individuals in a population at the beginning of a study or simulation. In the context of the video, the initial population is not explicitly known, but it is implied to be the value from which the geometric sequence begins. The script discusses using the population on day four as a proxy for the initial population to calculate the common ratio.
💡rate of change
The rate of change in a geometric sequence is represented by the common ratio, which determines how much each term in the sequence increases or decreases from the previous term. The script refers to this as 'R' and calculates it based on the known populations on day four and day eight, which is essential for predicting the population on day six.
💡nth root
The nth root of a number is a value that, when raised to the power of 'n', gives the original number. In the video, the script uses the fourth root to find the common ratio 'R' by taking the fourth root of the ratio of the populations on day eight to day four. This mathematical operation is key to determining the growth factor of the bacterial colony.
💡population
Population in the script refers to the number of bacteria in the colony at a given time. The video's main theme revolves around calculating the population at different days using a geometric sequence. The script provides specific population numbers for day four (4,000) and day eight (49,000) and aims to find the population for day six.
💡day four
Day four is a specific time point in the simulation where the population of the bacterial colony is known to be 4,000. It serves as a reference point for calculating the common ratio and predicting the population on other days, such as day six.
💡day eight
Day eight is another specific time point in the simulation where the population is known to be 49,000. This information, along with the population on day four, is used to calculate the common ratio and understand the growth pattern of the bacterial colony.
💡day six
Day six is the target time point for which the video aims to predict the bacterial population. The script uses the known populations on day four and day eight to calculate the population on day six, demonstrating the application of the geometric sequence model.
💡calculator
In the context of the video, a calculator is a tool used to perform mathematical operations, such as division and taking the nth root, to solve for the common ratio and predict the population on day six. The script mentions using a calculator to perform these operations.
💡common ratio
The common ratio, denoted as 'R' in the script, is the factor by which the population increases or decreases in a geometric sequence. The script calculates this ratio using the populations on day four and day eight, which is essential for determining the population on day six.
💡model
In the script, the term 'model' refers to the mathematical representation of the bacterial colony's growth as a geometric sequence. The model is used to predict the population at different days, with the script focusing on applying this model to find the population on day six.
Highlights

The problem involves a geometric sequence model for a bacterial colony population simulation.

The population on day four is known to be 4,000 bacteria.

The population on day eight is known to be 49,000 bacteria.

The goal is to determine the population on day six.

Geometric sequence formula is P(T) = P0 * R^T, where T is time in days.

Day four is used as a reference point with P0 = 4,000.

Day eight population is used to calculate the growth rate R.

Calculation of R involves dividing 49,000 by 4,000.

The fourth root of 12.25 gives the value of R.

R is calculated to be √14/2.

Using the value of R, the model is reapplied for day six.

Day six calculation involves 4,000 multiplied by (√14/2)^2.

The population on day six is found to be 14,000.

The correct answer for the population on day six is option C.

The problem-solving process involves a clear step-by-step approach.

The use of a calculator is emphasized for solving the geometric sequence.

The solution requires understanding the properties of geometric sequences.

The final answer is derived logically from the given data and calculations.

Transcripts

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