Question

A company's total sales (in millions of dollars) t months from now are given by S(t) = sqrt t + 2. In your own words, find and interpret S(20) and S'(20). Use these results to estimate the total sales after 24 months and after 25 months.

Answer 1

Answer:

S(20) = $\sqrt{20+2}$ = 4.69 ; value of total sales after 20 months is 4.69 million dollars

S'(20) = $\dfrac{1}{\sqrt{20+2} }$ = 0.21 ; increase in the total sales after 20 months is 0.21 millions of dollars

S(24) = $\sqrt{24+2}$ = 5.10;  5.10  millions of dollars

S(25) = $\sqrt{25+2}$ = 5.20 ;  5.20  millions of dollars

Step-by-step explanation:

$S(t) = \sqrt{t+2}$

$S(t+h) = \sqrt{t+2+h}$

$S(t+h) - S(t)= \sqrt{t+2+h }- \sqrt{t+2}$

$\dfrac{S(t+h) - S(t)}{h}= \dfrac{\sqrt{t+2+h }- \sqrt{t+2}}{h}$

By rationalization:

$\dfrac{\sqrt{t+2+h }- \sqrt{t+2}}{h} * \dfrac{ {\sqrt{t+2+h }- \sqrt{t+2}} }{ {\sqrt{t+2+h }- \sqrt{t+2}}}$

$\\ \\ \dfrac{t+2+h-t-2}{h \sqrt{t+2+h }+ \sqrt{t+2} } = \dfrac {h}{\sqrt{t+2+h }+\sqrt{t+2} }$

= $\dfrac {1}{\sqrt{t+2+h }+\sqrt{t+2} }$

$\lim_{k \to \infty} \dfrac {1}{\sqrt{t+2+h }+\sqrt{t+2} } = \dfrac{1}{t+2}$

$S' (t) = \dfrac{1}{\sqrt{t+2} }$

S(20) = $\sqrt{20+2}$ = 4.69 ; value of total sales after 20 months is 4.69 million dollars

S'(20) = $\dfrac{1}{\sqrt{20+2} }$ = 0.21 ; increase in the total sales after 20 months is 0.21 millions of dollars

S(24) = $\sqrt{24+2}$ = 5.10;  5.10  millions of dollars

S(25) = $\sqrt{25+2}$ = 5.20 ;  5.20  millions of dollars