Monthly normal rainfall data x y for a particular city are

Monthly normal rainfall data (x , y) for a particular city are (2 , 6.68), (6 , 1.66), (8 , 2.83), where x represents time in months (with x = 1 representing January) and y represents rainfall in inches. Find the values of a, b, and c such that the equation y = ax^2 + bx + c models this data. According to the model, how much rain should the city expect during September?

Solution

we have three values of (x , y) (2, 6.68),(6 , 1.66),(8 , 2.83)

and we have equation which models this data

y = ax² + bx + c

it means these values satisfies this equation so

for (2 , 6.68) 6.68 = 4a + 2b + c .......(1)

for (6 , 1.66) 1.66 = 36a + 6b + c ..........(2)

for (8 , 2.83) 2.83 = 64a + 8b + c ..........(3)

eq(2) - eq(1) 32a + 4b = - 5.02

after solving 8a + b = - 1.255 ..........(4)

eq(3) - eq(2) 60a + 6b = - 3.85

after solving 10a + b = - 0.6416 .........(5)

eq(5) - eq(4) 2a = 0.6134

a = 0.3067

put value of a in eq(4)

8*(0.3067) + b = - 1.255

2.4536 + b = - 1.255

b = -1.255 - 2.4536

b = - 3.708

put value of a and b in eq(1)

6.68 = 4(0.3067) + 2(-3.708) + c

6.68 - 1.2268 + 7.416 = c

c = 12.86

so the final equation is

y = 0.3067x² - 3.708x + 12.86

rain expected in the city during september.

september is 9^th month so x=9

y = 0.3067 * 9² - 3.708 * 9 + 12.86

y = 24.8427 - 33.372 + 12.86

y = 4.3307

so 4.3307inches rainfall expected in september month.