SOLUTION: 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 Upper C equals at squar

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Question 1153986: 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 Upper C equals at squared plus bt plus 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.
​(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its 1962 level.

Year.
1962
1982
2002
CO2
319
340
368
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!
The table lists a readings for three years.
​(a) If the quadratic relationship between the carbon dioxide concentration and the year is expressed as ​, where equals corresponds to​ 1962, use a system of linear equations to determine the constants​ , , and​ , and give the equation.

Year.
1962
1982
2002
CO2





since equals corresponds to​ 1962, and is



................year 1982, , ,


.......solve for
.................eq.1

................year 2002, , ,




........eq.2

from eq.1 and eq.2 we have





go to
.................eq.1, plug in


your equation is:



​(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its 1962 level.
1962 level is
in years will double which is
.........sole for



using quadratic formula, we get

=> exact solution (only positive root, we will disregard negative solution)
-> approximately
years from 1962 is: year and the amount of carbon dioxide in the atmosphere will double from its 1962 level.


check:





Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!




where t=0 corresponds to 1962.

data:
   year   t     C
  ----------------
   1962   0    319
   1982  20    340
   2002  40    368

(a) determine the coefficients a, b, and c and write the equation
Equations:

  [1]
  [2]
  [3]

  [4] (from [1] and [2])
  [5] (from [1] and [3]

  [6] ([4], doubled)

  ([5]-[6])
  [7]

  (from [5] and [7])

ANSWERS:
a = 7/800
b =7/8
c = 319


(b) Find the year when C = 2(319) = 638



Solving algebraically is certainly possible, but very tedious.

Graph the function and the constant 638 on a graphing calculator and find where they intersect.

A graph:



The intersection of the two graphs, to the nearest whole number, is 147, indicating the year 1962+147 = 2109.

ANSWER: year 2109
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