function problem

thatsnotalion

im using the equation p(t) = L/1+Ce^-kt

the density was measured at days 0 and 10 and was found to be 2 and 194 respectively.when k = 0.74 i need to find L and C. can someone please show me how?

Prove It

MHF Helper
im using the equation p(t) = L/1+Ce^-kt

the density was measured at days 0 and 10 and was found to be 2 and 194 respectively.when k = 0.74 i need to find L and C. can someone please show me how?
$$\displaystyle p(t) = \frac{L}{1 + Ce^{-kt}}$$.

You are told that $$\displaystyle k = 0.74$$ and you are given two $$\displaystyle (t, p)$$ points.

So $$\displaystyle 2 = \frac{L}{1 + Ce^{-0.74(0)}}$$ and $$\displaystyle 194 = \frac{L}{1 + Ce^{-0.74(10)}}$$.

That means you have two equations in two unknowns:

$$\displaystyle 2 = \frac{L}{1 + C}$$

$$\displaystyle 194 = \frac{L}{1 + Ce^{-7.4}}$$.

Dividing the second equation by the first yields:

$$\displaystyle \frac{194}{2} = \frac{\frac{L}{1 + Ce^{-7.4}}}{\frac{L}{1 + C}}$$

$$\displaystyle 97 = \frac{1 + C}{1 + Ce^{-7.4}}$$

$$\displaystyle 97(1 + Ce^{-7.4}) = 1 + C$$

$$\displaystyle 97 + 97Ce^{-7.4} = 1 + C$$

$$\displaystyle 97Ce^{-7.4} - C = 1 - 97$$

$$\displaystyle C(97e^{-7.4} - 1) = -96$$

$$\displaystyle C = -\frac{96}{97e^{-7.4} - 1}$$

$$\displaystyle C = \frac{96}{1 - 97e^{-7.4}}$$.

Substituting back into the first equation:

$$\displaystyle 2 = \frac{L}{1 + C}$$

$$\displaystyle 2 = \frac{L}{1 + \frac{96}{1 - 97e^{-7.4}}}$$

$$\displaystyle 2 = \frac{L}{\frac{97 - 97e^{-7.4}}{1 - 97e^{-7.4}}}$$

$$\displaystyle 2 = \frac{L}{\frac{97(1 - e^{-7.4})}{1 - 97e^{-7.4}}}$$

$$\displaystyle L = \frac{194(1 - e^{-7.4})}{1 - 97e^{-7.4}}$$.

thatsnotalion

thatsnotalion

thankyou so much!

thatsnotalion

sorry...one more question. how would i determine lim p(t) of t'towards' infinity. (sorry i dont know how to use the symbols)?

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MHF Helper
sorry...one more question. how would i determine lim p(t) of t'towards' infinity. (sorry i dont know how to use the symbols)?
Start by writing the function with $$\displaystyle k, C, L$$ substituted. Then decide what happens to a negative exponential function as $$\displaystyle t \to \infty$$...

thatsnotalion

thatsnotalion

ok i got (Le^(kt))/(e^(kt)+c)

thanks for your help!