# Thread: Need help finding exponential equation

1. ## Need help finding exponential equation

A bacteria culture grows with constant relative growth rate. After 2 hours there are 400 bacteria and after 8 hours the count is 50,000.

(a) Find the initial population.

(b) Find an expression for the population after t hours.

(c) Find the rate of growth after 3 hours.

Please help me with this problem. Also explain your steps and show your work. Thank you.

2. Your population growth is given by:

$\displaystyle P(t) = xt^n$

according to the question:

$\displaystyle P(2)= x2^n = 400$........(I)

$\displaystyle P(8) = x8^n = 50,000$.....(II)

Divide (II)/(I) to find n, and then x

after that, you can do (a),(b), and (c)

3. Thanks for helping me. So I could you correctly, you mean divide it like this?

50000 = x8^n
400 = x2^n

50000/400 = (8/2)^n?

Don't we need to use the formula y(t) = P0e^kt in here?

4. Sorry I should have used that equation $\displaystyle y(t) = P_{0}e^{kt}$

for this, you first need to find k, your rate..

suppose $\displaystyle P_{0} = 400$

then you have:

$\displaystyle 50,000 = 400 e^{k(8-2)}$

using this, you can find k.....

After finding k, you can find $\displaystyle y_{0}$ using:

$\displaystyle 50,000 = y_{0}e^{8k}$

5. Ah I see thank you for your help. May I ask one last question, how would we solve (c) Find the rate of growth after 3 hours? Do we need to find the derivative of y(t) = 80e^(0.80472t) in order to solve c? Thanks.

6. Yes

Then you put t = 3

### relative growth rate formula calculus

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