How do you do this math problem?
The number of bacteria b on a Petri dish is given by `b(t) = 500e^0.04t` , where t is measured in minutes.
a. how many bacteria are there at t=0?
b. How many bacteria will there be when t=30 minutes?
c. When will the bacteria double?
For each portion of the question, we simply are plugging in values to the funtion b(t):
`b(t) = 500e^(0.04t)`
a) How many bacteria are there at t=0?
To find this, we simply substitute 0 for t
`b(0) = 500e^(0.04*0)`
To solve this, we just need to recognize that e^0 = 1:
`b(0)=500*1 = 500`
Therefore, we have 500 bacteria as our starting point.
b) How many bacteria will there be at t=30 minutes?
Again, we will simply substitute 30 for t in b(t):
`b(30) = 500e^(0.04*30)`
`b(30) = 500e^1.2`
Now, we simply evaluate the above expression on a calculator to get:
`b(30) = 1660`
Wow, so apparently we have 1660 bacteria in just half an hour. This is reasonable, though, based on the equation and the fact that bacteria multiply fairly quickly. Now, we'll move on to the last part.
c) When will the bacteria double?
This question requires our answer from the first question. Our starting point was 500 bacteria (at time t=0). To determine when the number of bacteria will double, we'll need to determine when the number of bacteria reaches 1000 (2*500). To do this, we'll substitute 1000 into the b(t) term in our function and we'll solve for t:
`b(t) = 500e^(0.04t)`
`1000 = 500e^(0.04t)`
Now, we can divide both sides by 500:
`1000/500 = 500/500e^(0.04t)`
`2 = e^(0.04t)`
Now, to bring the exponent down and allow us to find t directly, we'll need to take the natural logarithm of both sides:
`ln(2) = ln(e^(0.04t))`
Recall, now, the property of natural logarithm that `ln(e^x) = x` because the natural logarithm is the inverse function of `e^x` (see link).``
In our case, we'll use this property to bring down the 0.04t:
`ln 2 =0.04t`
Now, we'll simply divide both sides by 0.04t:
`ln2/0.04 = (0.04t)/0.04`
`ln2/0.04 = t`
Now, depending on your teacher, you can leave it like that or just solve using a calculator to get the following result:
t = 17.3 minutes
Hope that helps!