Question 1154243

The table lists a readings for three years.

(a) If the quadratic relationship between the carbon dioxide concentration {{{C}}} and the year {{{t }}}is expressed as {{{C=at^2+bt+c}}}, where {{{t }}}equals {{{0}}} corresponds to {{{1962}}}, use a system of linear equations to determine the constants{{{a}}}, {{{b}}}, and {{{c}}}, and give the equation.

table:

{{{Year}}}|{{{CO2}}}{{{level}}}
{{{1962}}}|{{{318}}}
{{{1992}}}|{{{341}}}
{{{2022}}}|{{{371}}}



{{{C=at^2+bt+c}}}

 since {{{t}}} equals {{{0}}} corresponds to {{{1962}}}, and {{{C}}}  is {{{318}}}

{{{318=a*0^2+b*0+c}}}
{{{highlight(c=318)}}}


{{{C=at^2+bt+c}}}................year {{{1992}}}, {{{C=341}}}, {{{t=30}}}, {{{c=318}}}
{{{341=a*30^2+b*30+318}}}
{{{900a+30b=341-318}}}
{{{900a+30b=23}}}.......solve for {{{b}}}

{{{b=23/30-900a/30}}}

{{{b=23/30-30a}}}.................eq.1


{{{C=at^2+bt+c}}}................year {{{2022}}}, {{{C=371}}}, {{{t=60}}}, {{{c=318}}}
{{{371=a*60^2+b*60+318}}}
{{{3600a+60b=371-318}}}
{{{3600a+60b=53}}}
{{{60b=53-3600a}}}
 {{{b= 53/60-60a}}}........eq.2


from eq.1 and eq.2 we have

{{{23/30-30a=53/60-60a}}}

{{{30a+60a=53/60-23/30}}}

{{{90a=7/60}}}

{{{highlight(a=7/5400)}}}


go to

{{{b=23/30-30a}}}.................eq.1, plug in{{{ a}}}

{{{b=23/30-30(7/5400)}}}

{{{highlight(b=131/180)}}}

your equation is:

{{{highlight(C=(7/5400)t^2+(131/180)t+318)}}}



(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its {{{1962}}} level.

{{{1962}}} level is {{{318}}}
in {{{t }}}years will  double which is  {{{C=2*318=636}}}

{{{636=(7/5400)t^2+(131/180)t+318}}}.........sole for {{{t}}}

{{{(7/5400)t^2+(131/180)t+318-636=0}}}

{{{(7/5400)t^2+(131/180)t-319=0}}}


using quadratic formula, we get


{{{t = -1965/7 - (15 sqrt(70753))/7}}}=> we need only positive root, we will disregard negative solution 


{{{t = (15 sqrt(70753))/7 - 1965/7}}}-> exact solution

{{{t = 288.6}}}-> approximately


 {{{288.6}}} years from {{{1962}}} is:  the year {{{2250}}} and the amount of carbon dioxide in the atmosphere will double from its {{{1962}}} level.