The number of bacteria in a culture is modeled by n(t)=1550e^(0.24t)
(a) The initial number of bacteria is _____
(b) The relative rate of growth of this bacterium population is _____
(c) The number of bacteria after 3 hours is _____
(d) After how many hours will the number of bacteria reach 10,000? _____
(a) n(t) = 1550 (when t = 0)
(b) relative to what?
(c) If t is in hours in the equation, plug in t = 3 and calculate
(d) Solve 10,000 = 1550 e^0.24t for t
0.24 t = ln6.452 = 1.864
t = ____
(a) The initial number of bacteria can be found by plugging in t=0 into the equation:
n(0) = 1550e^(0.24*0) = 1550e^0 = 1550.
Therefore, the initial number of bacteria is 1550.
(b) The relative rate of growth of this bacterium population can be determined from the exponential growth equation. The coefficient of the exponent, in this case 0.24, represents the relative rate of growth.
Therefore, the relative rate of growth of this bacterium population is 0.24.
(c) To find the number of bacteria after 3 hours, we can plug t=3 into the equation:
n(3) = 1550e^(0.24*3) = 1550e^0.72 ≈ 3929.76.
Therefore, the number of bacteria after 3 hours is approximately 3929.76.
(d) To find the number of hours it takes for the number of bacteria to reach 10,000, we can set n(t) equal to 10,000 and solve for t:
10,000 = 1550e^(0.24t).
Dividing both sides by 1550:
e^(0.24t) ≈ 6.45161.
Taking the natural logarithm of both sides:
0.24t ≈ ln(6.45161).
Dividing both sides by 0.24:
t ≈ ln(6.45161)/0.24 ≈ 5. Assuming rounding to the nearest whole number, the number of hours it will take for the number of bacteria to reach 10,000 is approximately 5.
To find the answers to these questions, we'll use the given bacteria growth model equation: n(t) = 1550e^(0.24t)
(a) The initial number of bacteria is represented by n(0), so we substitute t = 0 into the equation:
n(0) = 1550e^(0.24 * 0) = 1550e^0 = 1550 * 1 = 1550
Therefore, the initial number of bacteria is 1550.
(b) The relative rate of growth is the coefficient value of the exponent in the equation. Here, it is 0.24.
Therefore, the relative rate of growth of this bacterium population is 0.24.
(c) To find the number of bacteria after 3 hours, substitute t = 3 into the equation:
n(3) = 1550e^(0.24 * 3) = 1550e^(0.72)
You can use a calculator to find the value of e^(0.72) - which is approximately 2.053.
n(3) ≈ 1550 * 2.053 ≈ 3180.515
Therefore, the number of bacteria after 3 hours is approximately 3180.515.
(d) To find the number of hours it takes for the number of bacteria to reach 10,000, we need to solve the equation n(t) = 10,000.
10,000 = 1550e^(0.24t)
Divide both sides by 1550:
e^(0.24t) ≈ 10,000/1550 ≈ 6.4516
Take the natural logarithm (ln) of both sides to isolate the exponent:
ln(e^(0.24t)) ≈ ln(6.4516)
0.24t ≈ ln(6.4516)
Divide both sides by 0.24:
t ≈ ln(6.4516) / 0.24
Using a calculator, t ≈ 4.1382
Therefore, after approximately 4.1382 hours, the number of bacteria will reach 10,000.
So, the complete answers are:
(a) The initial number of bacteria is 1550.
(b) The relative rate of growth of this bacterium population is 0.24.
(c) The number of bacteria after 3 hours is approximately 3180.515.
(d) After approximately 4.1382 hours, the number of bacteria will reach 10,000.