A culture starts with 8300 bacteria. After one hour the count is 10,000.
(a) Find a function that models the number of bacteria n(t) after t hours. (Round your r value to three decimal places.)
Why did the bacteria join a band? Because they had great "counting skills"!
Now, let's find the function that models the number of bacteria after t hours. We know that the initial count is 8300 bacteria, and after one hour it increases to 10,000.
This can be represented by the equation n(t) = a * b^t, where t is the number of hours and a and b are constants.
Substituting the given values, we have:
10,000 = a * b^1
Dividing both sides by 8300, we get:
10,000/8300 = a * b
Simplifying, we find:
1.205 = a * b
So the function that models the number of bacteria after t hours is:
n(t) = 1.205 * b^t.
To find a function that models the number of bacteria, we can use the exponential growth formula:
n(t) = n0 * e^(rt)
Where:
n(t) is the number of bacteria after t hours.
n0 is the initial number of bacteria.
e is the mathematical constant approximately equal to 2.71828.
r is the growth rate.
In this case, the initial number of bacteria is 8300. After one hour, the count is 10,000. Therefore, n0 = 8300 and n(1) = 10000.
Substituting these values into the formula:
10000 = 8300 * e^(r * 1)
We can divide both sides of the equation by 8300 to isolate the exponential term:
e^(r * 1) = 10000 / 8300
Taking the natural logarithm (ln) of both sides of the equation will help us solve for r:
ln(e^(r * 1)) = ln(10000 / 8300)
Using the logarithm property, ln(e^(r * 1)) simplifies to r * 1:
r = ln(10000 / 8300)
Evaluating this expression using a calculator, we find that r is approximately 0.189.
Therefore, the function that models the number of bacteria is:
n(t) = 8300 * e^(0.189t)
(Note: The value of r is rounded to three decimal places.)
To find a function that models the number of bacteria n(t) after t hours, we can use exponential growth since the number of bacteria is increasing over time.
The general form for exponential growth is given by the equation:
n(t) = n₀ * e^(rt)
where:
- n(t) is the number of bacteria after t hours
- n₀ is the initial number of bacteria
- e is Euler's number (approximately 2.71828)
- r is the growth rate (which we need to find)
In this case, the initial number of bacteria (n₀) is 8300, and after one hour (t=1), the count is 10,000.
Substituting these values into the exponential growth equation, we have:
10000 = 8300 * e^(r * 1)
Simplifying the equation, we get:
e^r = 10000/8300
To solve for r, we can take the natural logarithm (ln) of both sides of the equation:
ln(e^r) = ln(10000/8300)
Using the property of logarithms (ln(e^x) = x), the equation becomes:
r = ln(10000/8300)
Calculating the value of r using a calculator, we have:
r ≈ ln(10000/8300) ≈ 0.194
Therefore, the function that models the number of bacteria n(t) after t hours is:
n(t) = 8300 * e^(0.194t)
(rounding the growth rate, r, to three decimal places)
10000/8300 = 1.205
So,
n(t) = 8300 * 1.205^n