Question 1196927
{{{E(t) =0.2t + 135}}} if {{{6 <= t <= 15}}}

{{{E(t) = -0.2t + 141}}} if {{{15 < t ≤ 20


(a)

{{{t=10}}}  belongs to the interval  {{{6 <= t <= 15}}} so we use the first formula to evaluate {{{E(10) }}}

{{{E(10) =0.2*10 + 135 =137}}}

{{{t=16}}}  belongs to the interval {{{15 < t <= 20}}} so we use the second formula to evaluate {{{E(16)}}}

{{{E(16) =  -0.2*16+ 141=137.8}}}

{{{t=20}}}  belongs to the interval {{{15 < t <= 20}}} so we use the second formula to evaluate {{{E(20)}}}

{{{E(20) = -0.2*20+ 141=137}}}


since starts with the end of December {{{2002}}} and time {{{t}}} in {{{months}}}, if {{{t=10 }}} it  means {{{10 }}}months from the end of December {{{2002}}} will be October {{{2003}}}
and the Conference Board Index was{{{ 137}}}


if {{{t=16 }}} it  means 16 months ({{{1}}} year and {{{4}}} months) from the end of December {{{2002}}} will be April {{{2004}}}
and the Conference Board Index was{{{ 137.8}}}


if {{{t=20 }}} means {{{20 }}}months ({{{1}}} year and {{{8}}} months) from the end of December {{{2002}}} will be August {{{2004}}}
and the Conference Board Index was {{{137}}}



(b)
  

Sketch the graph of E. Use your graph to estimate when the index first reached 138.

<a href="https://ibb.co/HqC91s6"><img src="https://i.ibb.co/HqC91s6/download.png" alt="download" border="0"></a>


{{{ drawing( 600, 600, -20, 20, -20, 200,

green(line(6,136.2, 15,138)),  blue(line(15,138, 20,137)), circle(15,138,.25), locate(15,138,max(15,138)),
graph( 600, 600, -20, 20, -20, 200, 0)) }}}

{{{t =15}}}
 



(c)

Use technology to generate a table of values for {{{E(t)}}} with
 
{{{t }}}| {{{6}}} | {{{7}}}| {{{8}}} |{{{9 }}}|{{{10}}}

use first formula: {{{E(t) =0.2t + 135}}}

{{{E(6) =0.2*6 + 135=136.2}}}
{{{E(7) =0.2*7 + 135=136.4}}}
{{{E(8) =0.2*8 + 135=136.6}}}
{{{E(9) =0.2*9 + 135=136.6}}}

your table is:

{{{t}}} ....| ...{{{6}}} ....| ...{{{7}}}.....|... {{{8}}} ....|...{{{9}}}.... |...{{{10}}} ..|
{{{E(t)}}}| {{{ 136.2}}}| {{{136.4}}} | {{{136.6}}} | {{{136.8 }}}| {{{137}}} | 


for second table use first  formula, E(t) =0.2t + 135, and do same way as I  did  above

{{{t}}}|{{{ 11}}}|{{{ 12}}}|{{{ 13}}}|{{{ 14}}}|{{{ 15}}}
{{{E(t)}}}| {{{ .....}}}| {{{.......}}} | {{{.....}}} | {{{....}}}| {{{.....}}} | 

for third table use second  formula, {{{E(t) = -0.2t + 141}}}, and do same way as I  did  above


{{{t}}}|{{{ 16}}}|{{{ 17}}}|{{{ 18}}}|{{{ 19}}}|{{{ 20}}}
{{{E(t)}}}| {{{ .....}}}| {{{.......}}} | {{{.....}}} | {{{....}}}| {{{.....}}} |