A owner of a rapidly expanding business finds that the first five months of the year the sales (in thousands) are $4.0.$4.4,$5.2, $6.4, and $8.0. the owner plots these figures on a graph and conjunctures that for the rest of the year, the sales curve can be approximated by a quadratic polynomial. Find the least squares quadratic polynomial fit the sales curve, and project the sales for the twelve month of the year.
Answers
Given : first five months of the year the sales (in thousands) are $4.0.$4.4,$5.2, $6.4, and $8.0 of a rapidly expanding business
To find : quadratic polynomial fitting the sales curve and sales for the twelve month of the year
Solution:
Let say
f(x) = ax² + bx + c
f(1) = 4
=> 4 = a + b + c
f(2) = 4.4
=> 4.4 = 4a + 2b + c
f(3) = 5.2
=> 5.2 = 9a + 3b + c
3a + b = 0.4
5a + b = 0.8
=> 2a = 0.4
=> a = 0.2
b = -0.2
a + b + c = 4
=> c = 4
=> f(x) = 0.2x² - 0.2x + 4
Lets check for f(4) & f(5)
f(4) = 0.2 * 4² - 0.2*4 + 4 = 6.4
f(5) = 0.2 * 5² - 0.2*5 + 4 = 8
f(12) = 0.2 * 12² - 0.2*12 + 4 = 30.4 $
f(x) = 0.2x² - 0.2x + 4 fit the sale curve
30.4 $ sale in 12th Month
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