Q. 56 - Calculus

Question

$$\begin{gathered} \mathrm{Suppose} 100 \mathrm{mg} \mathrm{of} \mathrm{a} \mathrm{drug} \mathrm{is} \mathrm{administered} \mathrm{to} \mathrm{a} \mathrm{patient} \mathrm{each} \mathrm{morning} \mathrm{in} \mathrm{pill} \mathrm{form} \\ \mathrm{and} \mathrm{it} \mathrm{is} \mathrm{known} \mathrm{that} \mathrm{after} 24 \mathrm{hours} \mathrm{the} \mathrm{body} \mathrm{processes} 80\% \mathrm{of} \mathrm{the} \mathrm{drug} \mathrm{from} \mathrm{such} \mathrm{a} \mathrm{pill}, \\ \mathrm{leaving} 20\% \mathrm{of} \mathrm{the} \mathrm{drug} \mathrm{in} \mathrm{the} \mathrm{body}. \mathrm{The} \mathrm{amount} \mathrm{of} \mathrm{the} \mathrm{drug} \mathrm{in} \mathrm{the} \mathrm{body} \mathrm{right} \mathrm{after} \mathrm{the} \mathrm{first} \mathrm{pill} \mathrm{is} \mathrm{taken} \mathrm{is} \\ A\left(1\right) = 100 mg. 24 \mathrm{hours} \mathrm{later}, \mathrm{after} \mathrm{the} \mathrm{second} \mathrm{pill} \mathrm{has} \mathrm{been} \mathrm{taken}, \mathrm{the} \mathrm{amount} \mathrm{in} \mathrm{the} \mathrm{body} \mathrm{is} \\ A\left(2\right) = 100(0.2) + 100 = 120 mg. 48 \mathrm{hours} \mathrm{later}, \mathrm{the} \mathrm{amount} \mathrm{in} \mathrm{the} \mathrm{body} \mathrm{right} \mathrm{after} \mathrm{taking} \mathrm{the} \mathrm{third} \mathrm{pill} \mathrm{is} \\ A\left(3\right) = 100(0.2)(0.2) + 100(0.2) + 100 = 124 mg. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{a}\right) \mathrm{Explain} \mathrm{the} \mathrm{calculations} \mathrm{for} A\left(1\right), A\left(2\right), \mathrm{and} A\left(3\right) \mathrm{described} \mathrm{in} \mathrm{the} \mathrm{exercise}. \\ \mathrm{Which} \mathrm{term} \mathrm{in} A\left(3\right) \mathrm{corresponds} \mathrm{to} \mathrm{the} \mathrm{drug} \mathrm{left} \mathrm{from} \mathrm{the} \mathrm{first} \mathrm{pill}? \\ \left(\mathrm{b}\right) \mathrm{Interpret} \mathrm{the} \mathrm{given} \mathrm{graph} \mathrm{in} \mathrm{the} \mathrm{context} \mathrm{of} \mathrm{this} \mathrm{problem}. \mathrm{What} \mathrm{do} \mathrm{the} \mathrm{marked} \mathrm{points} \mathrm{represent}? \\ \left(\mathrm{c}\right) \mathrm{Express} A\left(n\right) \mathrm{in} \mathrm{sigma} \mathrm{notation}. \\ \left(\mathrm{d}\right) \mathrm{Calculate} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{drug} \mathrm{in} \mathrm{the} \mathrm{body} \mathrm{after} \mathrm{the} 4^{th} \mathrm{through} {10}^{th} \mathrm{pills}. \\ \mathrm{Do} \mathrm{you} \mathrm{notice} \mathrm{anything} \mathrm{special} \mathrm{about} A\left(n\right) \mathrm{as} \mathrm{n} \mathrm{gets} \mathrm{larger}? \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right). T\mathrm{he} \mathrm{term} \mathrm{is} :100(0.2) (0.2) \\ \left(b\right). T\mathrm{he} \mathrm{points} \mathrm{shown} \mathrm{below} \mathrm{in} \mathrm{the} \mathrm{graph} \mathrm{represents} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{drug} \mathrm{remains} \mathrm{from} \mathrm{the} \mathrm{previous} \mathrm{pills}. \\ \left(c\right). \sum _{k=1}^{n}100\left({\left(0.2\right)}^n+1\right) \\ \left(d\right). A\left(4\right) = 124.8 mg, A\left(5\right) = 124.96 mg, A\left(6\right) = 124.992 mg,..,A\left(10\right) = 124.9999872 mg \end{gathered}$$

Step 1. Given Information

$$\begin{gathered} 100\mathrm{mg} \mathrm{of} \mathrm{drug} \mathrm{is} \mathrm{administered} \mathrm{to} \mathrm{a} \mathrm{patient} \mathrm{each} \mathrm{morning} \mathrm{in} \mathrm{pill} \mathrm{form}, \\ \mathrm{after} 24 \mathrm{hours} \mathrm{the} \mathrm{body} \mathrm{processes} 80\% \mathrm{of} \mathrm{the} \mathrm{drug} \mathrm{leaving} 20\% \mathrm{in} \mathrm{the} \mathrm{body}. \\ \mathrm{Amount} \mathrm{of} \mathrm{Drug} \mathrm{in} 1\mathrm{st} \mathrm{day} \\ A\left(1\right) = 100mg \\ \mathrm{After} 24 \mathrm{hours}, \mathrm{the} \mathrm{amount} \mathrm{in} \mathrm{the} \mathrm{second} \mathrm{day} \mathrm{is}, \\ A\left(2\right) = 100(0.2) + 100 = 120mg \\ \mathrm{On} \mathrm{third} \mathrm{day}, \mathrm{the} \mathrm{amount} \mathrm{is} \\ A\left(3\right) = 100(0.2)(0.2) + 100(0.2) + 100 = 124mg. \end{gathered}$$

Step 2. Solution (a) : The objective is to explain the calculations for A(1), A(2), A(3) and then to determine the term in A(3) corresponding to the drug left from the first pill.

$$\begin{gathered} \mathrm{The} \mathrm{amount} \mathrm{of} \mathrm{drug} \mathrm{left} \mathrm{after} \mathrm{consuming} \mathrm{the} \mathrm{first} \mathrm{pill} \mathrm{is} 20\%=0.2 \mathrm{of} \mathrm{the} \\ \mathrm{amount} \mathrm{contained} \mathrm{in} \mathrm{the} 100 \mathrm{mg} \mathrm{of} \mathrm{the} \mathrm{drug}. \\ \mathrm{So}, A\left(1\right)=100 \\ A\left(2\right)=100(0.2)+100= 120mg \\ \mathrm{The} \mathrm{term} \mathrm{in} A\left(3\right) \mathrm{corresponding} \mathrm{to} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{drug} \mathrm{left} \mathrm{from} \mathrm{the} \mathrm{first} \mathrm{pill} \mathrm{is}, 100(0.2) (0.2) \\ \mathrm{Therefore}, \mathrm{the} \mathrm{term} \mathrm{is} :100(0.2) (0.2) \end{gathered}$$

Step 3. Solution (b) : The objective is to interpret the graph in the content of the problem and to determine what the marked points represents.

$$\begin{gathered} \mathrm{The} \mathrm{graph} \mathrm{represents} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{dose} \mathrm{remains} \mathrm{in} \mathrm{the} \mathrm{body} \mathrm{each} \mathrm{day} \mathrm{after} \mathrm{consuming} \mathrm{the} \mathrm{pills}. \\ \mathrm{The} \mathrm{marked} \mathrm{points} \mathrm{at} 100 \mathrm{represents} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{drug} \mathrm{contained} \mathrm{by} \mathrm{the} \mathrm{pill}. \\ \mathrm{And}, \mathrm{the} \mathrm{points} \mathrm{shown} \mathrm{below} \mathrm{in} \mathrm{the} \mathrm{graph} \mathrm{represents} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{drug} \mathrm{remains} \mathrm{from} \mathrm{the} \mathrm{previous} \mathrm{pills}. \end{gathered}$$

Step 4. Solution (c) : The objective is to express A(n) in sigma notation.

$$\begin{gathered} \mathrm{The} \mathrm{given} \mathrm{terms} \mathrm{are}, \\ A\left(1\right) = 100mg \\ A\left(2\right) = 100(0.2) + 100 = 120mg \\ A\left(3\right) = 100(0.2)(0.2) + 100(0.2) + 100 = 124mg \\ \mathrm{Therefore}, \mathrm{the} \mathrm{sigma} \mathrm{notation} \mathrm{is}: \\ \sum _{k=1}^{n}100\left({\left(0.2\right)}^n+1\right) \end{gathered}$$

Step 5. Solution (d) : Objective is find A(4), A(5),...,A(10) and determine the relation between A(n) and n.

$$\begin{gathered} \mathrm{Using} \mathrm{sigma} \mathrm{notation}: \\ A\left(4\right) = 124.8 mg \\ A\left(5\right) = 124.96 mg \\ A\left(6\right) = 124.992 mg \\ . \\ . \\ A\left(10\right) = 124.9999872 mg \\ \mathrm{On} \mathrm{observing} \mathrm{we} \mathrm{find} \mathrm{that} on \mathrm{increasing} \mathrm{n} \mathrm{after} 3, A\left(n\right) \mathrm{doesnt} \mathrm{increase} \mathrm{significantly}. \end{gathered}$$