Question

The number of bacteria in a culture grew from 275 to 1135 in three hours. How do you find the number of bacteria after 7 hours?

Answer 1

`7381`

Explanation:

Bacteria undergo asexual reproduction at an exponential rate. We model this behavior using the exponential growth function.

`color(white)(aaaaaaaaaaaaaaaaaa)color(blue)(y(t) = A_(o)*e^(kt)`

Where

• `"y("t") = value at time ("t")"`
• `A_("o") = "original value"`
• `"e = Euler's number 2.718"`
• `"k = rate of growth"`
• `"t = time elapsed"`

You are told that a culture of bacteria grew from `color(red)[275` to `color(red)[1135` in `color(red)"3 hours"`. This should automatically tell you that:

• `color(blue)[A_("o")` = `color(red)[275]`

• `color(blue)["y"("t")]` = `color(red)["1135"]`, and

• `color(blue)"t"` = `color(red)["3 hours"]`

Let's plug all this into our function.

`color(white)(aaaaaaaaaa)color(blue)(y(t) = A_(o)*e^(kt)) -> color(red)1135 = (color(red)275)*e^(k*color(red)3)`

We can work with what we have above because we know every value except for the `"rate of growth", color(blue)[k]"`, for which we will solve.

`color(white)(--)`

`ul"Solving for k"`

• `color(red)1135 = (color(red)275)*e^(k*color(red)3)`

• `stackrel"4.13"cancel[((1135))/((275))] = cancel[(275)/(275)]e^(k*3)`

• `4.13 = e^(k*3)`

• `color(white)(a)_(ln)4.13 = color(white)(a)_cancel(ln)(cancele^(k*3))`

• `1.42 = k*3`

• `stackrel"0.47"cancel[((1.42))/((3))] = k*cancel[(3)/(3)`

• `0.47 = k`

Why did we figure all this out? Didn't the question ask to solve for the number of bacteria after `"time = 7 hours"` and not for `color(blue)[k], "the rate of growth"`?

The simple answer is that we needed to figure out the `"rate of growth"` so that from there we can figure out the value at time `(7)` by setting up a new function since we will have only 1 unknown left to solve.
`color(white)(--)`

`ul"Solving for number of bacteria after 7 hours"`

`color(blue)(y(t) = A_(o)*e^(kt)) -> y = (275)*e^(0.47*7)`

`y = (275)*e^(3.29)`

`y = (275)*(26.84)`

`y = 7381`

So, the bacteria colony will grow to `7381` in number after `"7 hours"`