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atroni [7]
2 years ago
15

A quantity with an initial value of 8500 decays exponentially at a rate such that the

Mathematics
1 answer:
Vlada [557]2 years ago
7 0

Answer:

4250 = 8500 e^{50 k}

And we can divide both sides by 8500 and we got:

\frac{1}{2} = e^{50 k}

And we can solve for k using natural log

k = \frac{ln(\frac{1}{2})}{50}= -0.0138629

And we have the model give by:

P(t) = 8500 e^{-0.01386294361 t}

And if we replace t =97 we got:

P(t=97) = 8500 e^{-0.01386294361 *97}= 2215.240

Step-by-step explanation:

For this case we can use the proportional model given by:

\frac{dP}{dt}= kP

And we can rewrite the expression like this:

\frac{dP}{P} = k dt

If we integrate both sides we got:

ln P = kt +C

If we apply exponentials in both sides we got:

P = e^{kt +C}

And we can rewrite the expression like this:

P(t) = P_o e^{kt}

Where P represent the population and t the time in years since the starting value

For this case we have that P_o = 8500

And after t = 5*10 = 50 years the value is the half or 8500/2 = 4250

So we can use this condition and we have:

4250 = 8500 e^{50 k}

And we can divide both sides by 8500 and we got:

\frac{1}{2} = e^{50 k}

And we can solve for k using natural log

k = \frac{ln(\frac{1}{2})}{50}= -0.0138629

And we have the model given by:

P(t) = 8500 e^{-0.01386294361 t}

And if we replace t =97 we got:

P(t=97) = 8500 e^{-0.01386294361 *97}= 2215.240

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