The number of bacteria present in a culture is given by the following model B=1000e^0.276t; where t= time in minutes. Find the time required to have 6000 bacteria present. Then, round your answer to the nearest tenth of a minute.
just solve for t in
1000e^0.276t = 6000
e^.276t = 6
.276t = ln6
t = ln6/0.276
To find the time required to have 6000 bacteria present, we need to solve the equation B = 6000 for t.
Given equation: B = 1000e^0.276t
Substitute B with 6000 in the equation:
6000 = 1000e^0.276t
Divide both sides of the equation by 1000:
6 = e^0.276t
Take the natural logarithm (ln) of both sides to eliminate the exponential:
ln(6) = ln(e^0.276t)
Simplify the right side of the equation using the logarithmic rule:
ln(6) = 0.276t
Divide both sides of the equation by 0.276:
t = ln(6) / 0.276
Using a calculator, evaluate ln(6) / 0.276:
t ≈ 5.717
Round the answer to the nearest tenth of a minute:
t ≈ 5.7 minutes
Therefore, it would take approximately 5.7 minutes to have 6000 bacteria present in the culture.
To find the time required to have 6000 bacteria present, we need to solve the equation B = 6000 for t. The equation for the number of bacteria in the culture is B = 1000e^0.276t.
Setting B equal to 6000, we have:
6000 = 1000e^0.276t
Now, we need to isolate the variable "t".
First, we divide both sides of the equation by 1000:
6 = e^0.276t
Next, we take the natural logarithm (ln) of both sides to eliminate the exponential:
ln(6) = ln(e^0.276t)
Using the property of logarithms that ln(e^x) = x, the equation simplifies to:
ln(6) = 0.276t
Now, divide both sides of the equation by 0.276:
t = ln(6)/0.276
Using a calculator, we can find that ln(6) is approximately 1.7918 and divide that by 0.276 to get:
t ≈ 1.7918 / 0.276 ≈ 6.4891
Rounding to the nearest tenth of a minute, the time required to have 6000 bacteria present is approximately 6.5 minutes.