A shopkeeper earns Rs. 380 in the first week, Rs. 660 in the second week and Rs. 860 in the third week. Earning is the quadratic
1 answer:
9514 1404 393
Answer:
e(w) = -40w^2 +400w +20
Step-by-step explanation:
A graphing calculator or spreadsheet can do the quadratic regression and help you find the formula. Here earnings (e) as a function of weeks (w) can be expressed by ...
e(w) = -40w^2 +400w +20
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<em>Further development</em>
There are a couple of other ways you can go at this. One is to write equations for a, b, c using the standard form quadratic.
aw^2 +bw +c = e
a +b +c = 380 . . . . . . for week 1
4a +2b +c = 660 . . . for week 2
9a +3b +c = 860 . . . for week 3
The solution of this system of equations will give the coefficients of the quadratic. Your graphing calculator or any of several web sites can solve these equations for you.
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Consider the differences between quadratic values for w = 1, 2, 3.
e2 -e1 = (4a +2b +c) -(a +b +c) = 3a +b
e3 -e2 = (9a +3b +c) -(4a +2b +c) = 5a +b
Then the second differences are ...
(e3 -e2) -(e2 -e1) = (5a +b) -(3a +b) = 2a
In the given set of numbers, the first differences are ...
660 -380 = 280 . . . . . . . call this d1
860 -660 = 200
And the second difference is ...
200 -280 = -80 . . . . . . . call this d2
From the above, we know that 2a = -80 ⇒ a = -40
and 3a +b = 280 = 3(-40) +b ⇒ b = 400
Finally, a+b+c = 380 = -40 +400 +c ⇒ c = 20
Then the generic solution is ...
a = d2/2
b = d1 -3a
c = e1 -a -b . . . . . where e1 is the first-week's earnings
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