Q. 85 - Calculus

Question

$$\begin{gathered} \mathrm{Linda} \mathrm{can} \mathrm{sell} 12 \mathrm{magazine} \mathrm{subscriptions} \mathrm{per} \mathrm{week} \mathrm{and} \mathrm{makes} \backslash (4.00 \\ \mathrm{for} \mathrm{each} \mathrm{magazine} \mathrm{subscription} \mathrm{she} \mathrm{sells}. \mathrm{Obviously} \mathrm{this} \mathrm{means} \mathrm{that} \\ \mathrm{Linda} \mathrm{will} \mathrm{make} \left(12\right)(\backslash )4.00) = \$48.00 \mathrm{per} \mathrm{week} \mathrm{from} \mathrm{magazine} \mathrm{subscriptions}. \\ \mathrm{Explain} \mathrm{this} \mathrm{result} \mathrm{mathematically}, \mathrm{using} \mathrm{mathematical} \mathrm{notation} \mathrm{and} \mathrm{the} \mathrm{chain} \mathrm{rule}. \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{In} \mathrm{mathematical} \mathrm{notations} \\ \mathrm{The} \mathrm{amount} \mathrm{she} \mathrm{make} \mathrm{per} \mathrm{week} \mathrm{is}, \\ \frac{d\mathrm{D}}{d\mathrm{t}}=\frac{d\mathrm{D}}{d\mathrm{m}}\times \frac{d\mathrm{m}}{d\mathrm{t}} \\ =4\times 12 \\ = 48. \end{gathered}$$

Step 1. Given Information

$$\begin{gathered} \mathrm{The} \mathrm{number} \mathrm{of} \mathrm{magazine} \mathrm{subscriptions} \mathrm{per} \mathrm{week} \mathrm{Linda} \mathrm{can} \mathrm{sell} \mathrm{is} 12 \mathrm{the} \mathrm{earning} \mathrm{is} \$4. \\ \mathrm{per} \mathrm{magazine} \mathrm{is} \frac{d\mathrm{D}}{d\mathrm{m}} = 4. \end{gathered}$$

Step 2. Solution

$$\begin{gathered} \mathrm{The} \mathrm{amount} \mathrm{she} \mathrm{make} \mathrm{per} \mathrm{week} = \left(12\right)(\$4) = \$48. \\ \mathrm{In} \mathrm{mathematical} \mathrm{notations} \\ \mathrm{The} \mathrm{amount} \mathrm{she} \mathrm{make} \mathrm{per} \mathrm{week} \mathrm{is}, \\ \frac{d\mathrm{D}}{d\mathrm{t}}=\frac{d\mathrm{D}}{d\mathrm{m}}\times \frac{d\mathrm{m}}{d\mathrm{t}} \\ =4\times 12 \\ = 48. \end{gathered}$$