1. A colony of bacteria grows exponentially according to the function where N is measured in grams and t is measured in days.
N(t)=100e^0.045t
You may use graphical method or algebraic method. Please show work if algebraic or show the graph if graphical method - round to 2 decimals.
a. Determine the initial amount of bacteria: ____________________________
b. What is the population after 5 days: _______________________________
c. How long will it take the population to reach 140 grams:_______________
N(t) = 100*e^0.045t.
a. N(0) = 100*e^0 = 100*1 = 100 g.
b. N(5) = 100*e^(0.045*5) = 100*e^0.225 = 125.2 g.
c. N = 100*e^0.045t = 140 g.
100*e^0.045t = 140.
e^0.045t = 140/100 = 1.40.
Take Ln of both sides:
0.045t*Lne = Ln1.40.
0.045t = Ln1.40 / Lne = 0.33647.
t = 0.33647 / 0.045 = 7.5 Days.
a. Determine the initial amount of bacteria:
To find the initial amount of bacteria, we can substitute t = 0 into the equation N(t) = 100e^(0.045t):
N(0) = 100e^(0.045*0)
N(0) = 100e^0
N(0) = 100(1)
N(0) = 100 grams
Therefore, the initial amount of bacteria is 100 grams.
b. What is the population after 5 days:
To find the population after 5 days, we can substitute t = 5 into the equation N(t) = 100e^(0.045t):
N(5) = 100e^(0.045*5)
N(5) = 100e^0.225
N(5) ≈ 100(1.253)
N(5) ≈ 125.3 grams
Therefore, the population after 5 days is approximately 125.3 grams.
c. How long will it take the population to reach 140 grams:
To determine how long it will take the population to reach 140 grams, we need to solve the equation N(t) = 140. Let's set up the equation:
140 = 100e^(0.045t)
To solve for t, we can divide both sides by 100 and take the natural logarithm of both sides:
ln(1.4) = ln(e^(0.045t))
ln(1.4) = 0.045t
t ≈ ln(1.4)/0.045
t ≈ 13.94 days
Therefore, it will take approximately 13.94 days for the population to reach 140 grams.
a. To determine the initial amount of bacteria, we need to find N(0), which represents the amount of bacteria at t = 0.
Substitute t = 0 into the equation:
N(0) = 100e^(0.045 * 0)
N(0) = 100e^0
N(0) = 100 * 1
N(0) = 100
Therefore, the initial amount of bacteria is 100 grams.
b. To find the population after 5 days, we need to find N(5) by substituting t = 5 into the equation:
N(5) = 100e^(0.045 * 5)
N(5) = 100e^0.225
N(5) = 100 * 1.252
N(5) ≈ 125.2
Therefore, the population after 5 days is approximately 125.2 grams.
c. To determine how long it takes for the population to reach 140 grams, we need to find the value of t when N(t) = 140.
Substitute N(t) = 140 into the equation:
140 = 100e^(0.045t)
Divide both sides by 100:
1.4 = e^(0.045t)
Take the natural logarithm (ln) of both sides to remove the exponential function:
ln(1.4) = ln(e^(0.045t))
Simplify the right side using the property of logarithms:
ln(1.4) = 0.045t
Divide both sides by 0.045 to solve for t:
t = ln(1.4) / 0.045 ≈ 18.89
Therefore, it will take approximately 18.89 days for the population to reach 140 grams.
To determine the initial amount of bacteria, we need to find the value of N(0) in the given bacterial growth equation N(t) = 100e^(0.045t).
a. Initial amount of bacteria:
Substituting t = 0 into the equation, we have:
N(0) = 100e^(0.045 * 0)
N(0) = 100e^0
N(0) = 100 * 1
N(0) = 100
Therefore, the initial amount of bacteria is 100 grams.
b. Population after 5 days:
To find the population after 5 days, we need to evaluate N(5) in the given equation:
N(5) = 100e^(0.045 * 5)
N(5) = 100e^0.225
Using a calculator or decimal approximation of the exponential expression, we find that N(5) ≈ 123.57 grams.
Therefore, the population after 5 days is approximately 123.57 grams.
c. Time to reach a population of 140 grams:
We need to solve the equation N(t) = 140 for t. By rearranging the equation, we have:
140 = 100e^(0.045t)
Dividing both sides by 100:
1.4 = e^(0.045t)
To isolate the exponential term, we take the natural logarithm (ln) of both sides:
ln(1.4) = ln(e^(0.045t))
Using the logarithmic property ln(e^x) = x:
ln(1.4) = 0.045t
Dividing both sides by 0.045:
t = ln(1.4) / 0.045
Using a calculator or decimal approximation, we find that t ≈ 6.25 days.
Therefore, it will take approximately 6.25 days for the population to reach 140 grams.