Question

A quantity P is an exponential function of time I, such that P = 160 when t = 6 and P = 150 when I = 4. Use the given information about the function P = Poel to:

(a) Find values for the parameters k and Po.
Round your answers to three decimal places.
k=
Po =

Answer 1

• k = 0.032
• P0 = 131.836

Explanation:

Perhaps you want to use the points (t, P) = (4, 150) and (6, 160) to find the parameters P0 and k in the equation ...

`P(t)=P_0\cdot e^(kt)`

We know from the given points that we can write the equation as ...

`P(t)=150\left((160)/(150)\right)^((t-4)/(6-4))=150\left((16)/(15)\right)^{(t)/(2)-2}\\\\=150\left((16)/(15)\right)^(-2)*\left(\left((16)/(15)\right)^{(1)/(2)}\right)^t`

Comparing this to the desired form, we see that ...

`P_0=150\left((16)/(15)\right)^(-2)\approx 131.836\\\\e^(k)=\left((16)/(15)\right)^(1/2)\rightarrow k=(1)/(2)(ln(16)-ln(15))\approx 0.0322693`

So, the approximate equation for P is ...

`P(t)=131.836\cdote^(0.032t)`

And the parameters of interest are ...

• k = 0.032
• P0 = 131.836