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) About Me  (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 C and the year t+is expressed as C=at%5E2%2Bbt%2Bc​, 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.

Year.
1962
1982
2002
CO2
319
340
368

C=at%5E2%2Bbt%2Bc
since t equals 0 corresponds to​ 1962, and C is 319
319=a%2A0%5E2%2Bb%2A0%2Bc
highlight%28c=319%29

C=at%5E2%2Bbt%2Bc................year 1982, C=340, t=20, c=319
340=a%2A20%5E2%2Bb%2A20%2B319
400a%2B20b=340-319
400a%2B20b=21.......solve for b
b=21%2F20-20a.................eq.1

C=at%5E2%2Bbt%2Bc................year 2002, C=368, t=40, c=319
368=a%2A40%5E2%2Bb%2A40%2B319
1600a%2B40b=368-319
1600a%2B40b=49
40b=49-1600a
b=+49%2F40-40a........eq.2

from eq.1 and eq.2 we have
21%2F20-20a=49%2F40-40a
40a-20a=49%2F40-21%2F20
20a=7%2F40
highlight%28a=7%2F800%29

go to
b=21%2F20-20a.................eq.1, plug in+a
b=21%2F20-20%287%2F800%29
highlight%28b=7%2F8%29
your equation is:
highlight%28C=%287%2F800%29t%5E2%2B%287%2F8%29t%2B319%29


​(b) Predict the year when the amount of carbon dioxide in the atmosphere will double from its 1962 level.
1962 level is 319
in t+years will double which is C=2%2A319=638
638=%287%2F800%29t%5E2%2B%287%2F8%29t%2B319.........sole for t
%287%2F800%29t%5E2%2B%287%2F8%29t%2B319-638=0
%287%2F800%29t%5E2%2B%287%2F8%29t-319=0

using quadratic formula, we get

t+=+30+sqrt%28303%2F7%29+-+50=> exact solution (only positive root, we will disregard negative solution)
t+=+147.38-> approximately
147.38 years from 1962 is: year 2109 and the amount of carbon dioxide in the atmosphere will double from its 1962 level.


check:
638=%287%2F800%29%28147.38%29%5E2%2B%287%2F8%29%28147.38%29%2B319
638=%287%2F800%29%28147.38%29%5E2%2B%287%2F8%29%28147.38%29%2B319
638=190.05%2B128.95%2B319
638=638

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


C+=+at%5E2%2Bbt%2Bc

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:

c+=+319  [1]
400a%2B20b%2Bc+=+340  [2]
1600a%2B40b%2Bc+=+368  [3]

400a%2B20b+=+21  [4] (from [1] and [2])
1600a%2B40b+=+49  [5] (from [1] and [3]

800a%2B40b+=+42  [6] ([4], doubled)

800a+=+7  ([5]-[6])
a+=+7%2F800  [7]

14%2B40b+=+49  (from [5] and [7])
40b+=+35
b+=+7%2F8

ANSWERS:
a = 7/800
b =7/8
c = 319
C+=+%287%2F800%29t%5E2%2B%287%2F8%29t%2B319

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

%287%2F800%29t%5E2%2B%287%2F8%29t%2B319+=+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