Question

Drug Dosage Problem

A patient takes a 30-mg (milligram) antibiotic capsule every hour. At the end of any one hour, the amount of antibiotic remaining in her body is only 90% of the amount at the beginning of that hour. Your objective is to predict the total amount in her body after many hours.

1. The first 30 mg dose is taken at time t=0 h. How much of this dose remains at the end of 1

h? 2 h?     3 h? 4 h?

2. Starting with the last dose, the amounts remaining from the first 3 doses (t=4h) can be written

            30, 30(0.9), 30(0.9)², 30(0.9)³, 30(0.9)⁴

These numbers are part of a geometric sequence. The next term in the sequence is formed by multiplying the preceding term by 0.9, the common ratio. The total amount in the patient’s body after these 5 doses is a partial sum of the geometric series,

30+ 30(0.9) + 30(0.9)²+30(0.9)³+30(0.9)⁴

How many milligrams of the antibiotic are in her body after these 5 doses?

3. You can use the SUM and SEQUENCE commands on your grapher to compute partial sums of any series. Write the appropriate commands to sum the formula 30(0.9)ⁿ from n=0 to n=t. Check your commands by showing that you get the same sum as in Problem 2 when t=4 h.

4. Find the 11^th partial sum (t=10) and the 21^st partial sum (t=20).

5. The partial sums for this series converge to a limit. What does this limit appear to be? What implications does the convergence of this series have for the patient in this problem?

Regular Savings Problem.

Ernest Lee Dunn puts $800 into his IRA each year. The money earns 10% per year interest (APR), which means that at the end of any year the amount in the IRA is 1.1 times the amount at the beginning of that year.

6. The amount in the account at the beginning of year t is the sum of the worths of each deposit. Starting with the worth of the last deposit, the total is a partial sum of this geometric series:

            800+ 800(1.1)+ 800(1.1)²+¼+ 800(1.1)^t

Find the totals in the account for t = 10 yr., 20 yr, and 30 yr.

7. Do the totals in Problem 6 seem to be converging to a finite limit or diverging toward infinity?

8. What do you know about partial sums of geometric series that you did not know before?

PLEASE DO 6 AND 7

Answer 1

#6

Since this form the geometric series .

800[1+1.1+(1.1)^2+(1.1)^3....(1.1)^t]

Therefore to find totals in the account for different values of t we use formula

here r=1.1

For t=10 we plugin n=10 and r=1.1 into this formula and first term (a)=800

For t=20 years

For t=30 years

#7

WE observe that totals are getting increased towards an infinite amount for infinite years.

Also we can check it by looking at common ratio r=1.1

Since r=1.1 which is greater than 1 so this series will diverge.

So all the totals will be diverging towards infinity