The rate of change would be 300 bacteria per minute at 0.1 minutes past 1 PM
The exponential function of the bacteria
The given parameters are:
Initial, a = 20
At 30 minutes, Value = 500
An exponential function is represented as:
y = ab^t
Where a is the initial value.
So, we have:
y = 20b^t
After 30 minutes, we have:
20b^30 = 500
Divide both sides by 20
b^30 = 25
Take the 30th root of both sides
b = 1.11
Substitute b = 1.11 in y = 20b^t
y = 20(1.11)^t
Hence, the exponential model is y = 20(1.11)^t
Minutes to reach 11,000
This means that:
y = 11,000
So, we have:
20(1.11)^t = 11000
Divide both sides by 20
1.11^t = 550
Take the logarithm of both sides
tlog(1.1) = log(550)
Divide both sides by log(1.1)
t = 66
Hence, the population would reach 11,000 after 66 minutes
The population at 2PM
This means that:
t = 60 i.e. 60 minutes after 1PM
So, we have:
y = 20(1.11)^60
Evaluate
y = 10481
Hence, the population at 2PM is 10481
The rate of change at 2PM
In (c), we have:
y = 10481 ------ population at 2PM is 10481
t = 60
The rate of change is calculated as:
Rate = 10481/60
Evaluate
Rate = 174.68
Hence, the rate of change at 2PM is 174.68 bacteria per minute.
When the rate would be 300 bacteria per minute?
The rate of change is calculated as:
Rate = Population/Time
So, we have:
y/t = 300
This gives
y = 300t
Substitute y = 300t in y = 20(1.11)^t
300t = 20(1.11)^t
Divide both sides by 20
15t = (1.11)^t
Using a graphing calculator, we have:
t = 0.1
Hence, the rate would be 300 bacteria per minute at 0.1 minutes past 1 PM
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