43–44. Periodic doses
Suppose you take a dose of m mg of...

Question

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is

Aₙ = m + mf + ⋯ + mfⁿ⁻¹.

For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.

43.f = 0.25,m = 200 mg

Accepted Answer

A machine injects the liters of coolant into a system every cycle. Let T be the fraction of coolant remaining after one cycle. Right after the end cycle, the total coolant is TN equals D plus DT plus D T squared, and so on, and so you get to DT raise the N minus 1 power. If D equals 50 L and T equals 0.8, evaluate the limit as an approaches infinity of T sub N. What does this limit signify? Now we have our given series T N. Now, we noticed something about the series. This is a geometric series. We can write this as T N equals D multiplied by the sum. SK equals 0. 2 and minus 1 Of tea raised to the cake. Where our first term is D and common ratio is T. We can now write the sum of our first in terms or our partial sum. Tip in Equals D multiplied. By 1 minus T raises the N divided by 1 minus T. Let's find the limit of this. The Lim as in the approaches infinity of T N equals the limit, as in the approaches infinity. Of D multiplied by 1 minus T to the N divided by 1 minus T. Now let's go ahead and evaluate. Our T rates to the N, as long as Ts between 0 and 1, should go to 0. So, we have D multiplied by 1 minus 0. Divided by 1 minus T, which is just D divided by 1 minus T. Now you can find the specific value using our known values. We said D Was 50 and T. Is 0.8. Which your rights are a limit. As it approaches infinity, T N. As 50 divided by 1 minus 0.8. This simplifies to give us a value of 250. Now, this is our limit, and this represents the steady state amounts. In other words, this just means after many cycles, the total amount of coolant stabilizes at 250 L. So, this is the answer to our problem. OK, I hope to help me solve the problem. Thank you for watching. Goodbye.

Key Concepts

Geometric Series

Limit of a Geometric Series

Interpretation of the Limit in Context

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