Exponential Growth Homework: Find y, k, and When Population Reaches 10K

In summary, the bacteria culture starts with 1000 bacteria and grows to a population of 7000 after an hour. To find the size of the population after t hours, we use the equation y = yoe^{kt}. By setting y = 7000 and yo = 1000, we can solve for k and get ln7/t = k. To find the size of the population after 3 hours, we simply plug in t = 3 and solve for y. To find the rate of growth after 3 hours, we take the derivative of the equation y = yoe^{kt} and plug in t = 3. Lastly, to find when the population will reach 10 000, we can use the
  • #1
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Homework Statement



A bacteria culture starts with 1000 bacteria and grows to a population of 7000 after an hour.

a) Find the size of the population after t hours.
b) Find the size of the population after 3 hours.
c) Find the rate of growth of the population after 3 hours.
d) When will the population reach 10 000?

Homework Equations



y = yoe[itex]^{kt}[/itex]

The Attempt at a Solution



a) y = yoe[itex]^{kt}[/itex]
7000 = 1000e[itex]^{kt}[/itex]
7 = e[itex]^{kt}[/itex]
ln7 = kt
ln7/t = k (is this it?)

b) y = yoe[itex]^{kt}[/itex]
y = 1000e[itex]^{k(2)}[/itex] (am I on the right track?)

c) I know I have to take the derivative but what do I take the derivative of?

d) No idea.
 
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  • #2
Incog said:
a) y = yoe[tex]^{kt}[/tex]
7000 = 1000e[tex]^{kt}[/tex]
7 = e[tex]^{kt}[/tex]
ln7 = kt
ln7/t = k (is this it?)

You're close. What is the value of t when y = 7000?
 
  • #3
IM havng a similar problem with N=Ae^(kt)

N is the number of bcteria in a culture at difrent times (t days) as expressed as a table

T= 0 ... 1 ... 2 ... 3 ... 4 ... 5 .
N= 10^3. 7.4x10^3. 5.5x10^4. 4x10^5. 3x10^6. 2.2x10^7.


it asks to envestigate the process with reference to the function above, I am confused and needs it explained
 
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1. What is exponential growth?

Exponential growth is a mathematical model that describes the rapid increase of a quantity over time. It is characterized by a constant rate of change, where the quantity grows at an increasing rate.

2. How do you find y and k in exponential growth?

In the equation y = k * e^(rt), y represents the final quantity, k represents the starting quantity, e is the mathematical constant approximately equal to 2.71828, r is the growth rate, and t is the time. To find y and k, you need to know the starting and final quantities, the growth rate, and the time. Plug these values into the equation and solve for y and k.

3. What does the value of k represent in exponential growth?

The value of k in exponential growth represents the starting quantity or initial value. It is the quantity at the beginning of the growth period before any growth has occurred.

4. How do you determine when a population reaches a certain size in exponential growth?

In exponential growth, the population size can be determined by the equation y = k * e^(rt). To determine when the population reaches a certain size, plug in the starting and final population sizes, the growth rate, and solve for t, which represents the time it takes for the population to reach the desired size.

5. What are the limitations of using exponential growth to model population growth?

While exponential growth is a useful mathematical model, it does not always accurately represent real-world situations. It assumes constant growth rate, which is not always the case in natural populations. Additionally, it does not take into account factors such as limited resources, competition, and environmental changes that can affect population growth.

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