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Explain why the situation you describe is an example of exponential growth.

An animal is increasing in size by 40 percent.

7 answers

Exponential growth refers to a situation where the rate of growth of a quantity is proportional to its current size. In the context of an animal increasing in size by 40 percent, this means that the increase in size is not constant but rather depends on the current size of the animal.

Here's how it relates to exponential growth:

  1. Percentage Growth: If an animal increases in size by 40%, this means that each time period (let's say one year), the increase is based on its size at the beginning of that period. For instance, if the animal initially weighs 100 units, a 40% increase in size would result in a new size of 140 units.

  2. Continuing Growth: In the next period, the animal will grow by 40% of its new size (140 units), which would add an additional 56 units, leading to a total size of 196 units. This shows how each growth period results in a larger absolute increase because the percentage is applied to a larger size.

  3. Mathematical Representation: The growth can be modeled mathematically as a function where \( S(t) \) represents the size at time \( t \), and the growth can be described by the formula: \[ S(t) = S(0) \times (1 + r)^t \] where \( S(0) \) is the initial size, \( r \) is the growth rate (in this case, 0.40 for 40%), and \( t \) is time.

  4. Feedback Loop: As the animal continues to grow, the amount of size it gains each time period increases, leading to a rapid rise in size as time progresses. This feedback loop is a hallmark of exponential growth, where increases tend to accelerate over time.

In summary, this situation outlines exponential growth because the increase in size is constantly proportional to the current size of the animal, leading to compounding growth effects over successive time periods.

the animal is 30 pounds what are ten terms

To illustrate the exponential growth of the animal that is initially 30 pounds and increases in size by 40% each time period, we can compute the size of the animal over the course of ten terms (time periods). The growth can be calculated using the formula:

\[ S(t) = S(0) \times (1 + r)^t \]

Where:

  • \( S(0) \) is the initial size (30 pounds).
  • \( r \) is the growth rate (0.40).
  • \( t \) is the number of time periods.

Now, let's calculate the size over ten terms:

  1. Term 0 (Initial size): \[ S(0) = 30 \text{ pounds} \]

  2. Term 1: \[ S(1) = 30 \times (1 + 0.40)^1 = 30 \times 1.40 = 42 \text{ pounds} \]

  3. Term 2: \[ S(2) = 30 \times (1 + 0.40)^2 = 30 \times 1.96 = 58.8 \text{ pounds} \]

  4. Term 3: \[ S(3) = 30 \times (1 + 0.40)^3 = 30 \times 2.744 = 82.32 \text{ pounds} \]

  5. Term 4: \[ S(4) = 30 \times (1 + 0.40)^4 = 30 \times 3.8416 = 115.25 \text{ pounds} \]

  6. Term 5: \[ S(5) = 30 \times (1 + 0.40)^5 = 30 \times 5.378 = 161.34 \text{ pounds} \]

  7. Term 6: \[ S(6) = 30 \times (1 + 0.40)^6 = 30 \times 7.529 = 225.87 \text{ pounds} \]

  8. Term 7: \[ S(7) = 30 \times (1 + 0.40)^7 = 30 \times 10.482 = 314.46 \text{ pounds} \]

  9. Term 8: \[ S(8) = 30 \times (1 + 0.40)^8 = 30 \times 14.696 = 440.88 \text{ pounds} \]

  10. Term 9: \[ S(9) = 30 \times (1 + 0.40)^9 = 30 \times 20.739 = 622.17 \text{ pounds} \]

  11. Term 10: \[ S(10) = 30 \times (1 + 0.40)^{10} = 30 \times 29.637 = 889.11 \text{ pounds} \]

Summary of sizes over ten terms:

  1. Term 0: 30 pounds
  2. Term 1: 42 pounds
  3. Term 2: 58.8 pounds
  4. Term 3: 82.32 pounds
  5. Term 4: 115.25 pounds
  6. Term 5: 161.34 pounds
  7. Term 6: 225.87 pounds
  8. Term 7: 314.46 pounds
  9. Term 8: 440.88 pounds
  10. Term 9: 622.17 pounds
  11. Term 10: 889.11 pounds

This progression clearly illustrates exponential growth, as the size of the animal increases rapidly over time due to the compounding effect of the 40% increase in size.

whats the formula

The formula for calculating the size of the animal after a certain number of time periods, given an initial size and a constant percentage growth rate, is:

\[ S(t) = S(0) \times (1 + r)^t \]

Where:

  • \( S(t) \) is the size at time \( t \).
  • \( S(0) \) is the initial size of the animal.
  • \( r \) is the growth rate expressed as a decimal (in your case, \( r = 0.40 \) for a 40% increase).
  • \( t \) is the number of time periods.

Explanation of Terms:

  • Initial Size (\( S(0) \)): This is the starting size of the animal. For your example, \( S(0) = 30 \) pounds.
  • Growth Rate (\( r \)): This represents how much the size increases in each time period as a percentage, expressed in decimal form (40% = 0.40).
  • Time Period (\( t \)): This is the number of intervals over which growth is calculated (e.g., years, months).

Using this formula allows you to calculate the size of the animal for any given time period \( t \) by plugging in the initial size, growth rate, and the number of periods.

Here’s how it looks for your situation with the example values:

  1. Initial size (\( S(0) \)) = 30 pounds
  2. Growth rate (\( r \)) = 0.40
  3. For different values of \( t \) (from 0 to 10), you can calculate \( S(t) \) directly.

Example Calculation:

To find the size after 5 periods: \[ S(5) = 30 \times (1 + 0.40)^5 = 30 \times (1.40)^5 \approx 30 \times 5.378 = 161.34 \text{ pounds} \]

Feel free to use the formula with any starting size or growth rate that you want!

numbers no symbols

Sure! Here is the formula without any symbols, just using words and numbers:

To calculate the size at a given time period, use the following steps:

  1. Start with the initial size of the animal, which is 30 pounds.
  2. Add 1 to the growth rate of 0.40 (which represents a 40% increase).
  3. Raise the result from the second step to the power of the number of time periods you want to calculate.
  4. Multiply the initial size by the result from step three.

So, the steps in a more detailed format without symbols would be:

  • Size at time t equals the initial size (which is 30) multiplied by 1.40 raised to the power of t.

Now, here are the size calculations for the first ten terms:

  1. For Term Zero (initial size):

    • Size equals 30 pounds.

  2. For Term One:

    • Size equals 30 multiplied by 1.40 raised to the power of 1, which equals 42 pounds.

  3. For Term Two:

    • Size equals 30 multiplied by 1.40 raised to the power of 2, which equals 58.8 pounds.

  4. For Term Three:

    • Size equals 30 multiplied by 1.40 raised to the power of 3, which equals 82.32 pounds.

  5. For Term Four:

    • Size equals 30 multiplied by 1.40 raised to the power of 4, which equals 115.25 pounds.

  6. For Term Five:

    • Size equals 30 multiplied by 1.40 raised to the power of 5, which equals 161.34 pounds.

  7. For Term Six:

    • Size equals 30 multiplied by 1.40 raised to the power of 6, which equals 225.87 pounds.

  8. For Term Seven:

    • Size equals 30 multiplied by 1.40 raised to the power of 7, which equals 314.46 pounds.

  9. For Term Eight:

    • Size equals 30 multiplied by 1.40 raised to the power of 8, which equals 440.88 pounds.

  10. For Term Nine:

    • Size equals 30 multiplied by 1.40 raised to the power of 9, which equals 622.17 pounds.

  11. For Term Ten:

    • Size equals 30 multiplied by 1.40 raised to the power of 10, which equals 889.11 pounds.

This gives you a clear understanding of how to calculate the size at each time period without using mathematical symbols.