SOLUTION: A company has sales (measured in millions of dollars) of 50, 60, and 75 during the first three consecutive years. Find a quadratic function that fits these data, and use the result

Question 310097: A company has sales (measured in millions of dollars) of 50, 60, and 75 during the first three consecutive years. Find a quadratic function that fits these data, and use the result to predict the sales during the fourth year. Assume that the quadratic function is of the form y = ax2 + bx + c
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A company has sales (measured in millions of dollars) of 50, 60, and 75 during the first three consecutive years.
Find a quadratic function that fits these data, and use the result to predict the sales during the fourth year.
Assume that the quadratic function is of the form y = ax2 + bx + c
:
Let x = no. of the year, we can develop 3 equations from this
:
Eq1: x=1, y=50
a(1^2) + 1b + c = 50
a + b + c = 50
:
Eq2: x=2, y=60
a(2^2) + 2b + c = 60
4a + 2b + c = 60
:
Eq3: x=3, y=75
a(3^2) + 3b + c = 75
9a + 3b + c = 75
:
Subtract eq1 from eq2
4a + 2b + c = 60
a + b + c = 50
--------------------Eliminates c
3a + b = 10
:
Subtract eq2 from eq3
9a + 3b + c = 75
4a + 2b + c = 60
---------------------eliminates c again
5a + b = 15
:
Using these two 2 unknown equation to eliminate b
5a + b = 15
3a + b = 10
----------------Subtraction eliminates b, find a
2a = 5
a = 2.5
:
Use the eq: 5a + b = 15, to find b
5(2.5) + b = 15
12.5 + b = 15
b = 15 - 12.5
b = 2.5
:
Use eq1 to find c
2.5 + 2.5 + c = 50
5 + c = 50
c = 50 - 5
c = 45
:
The equation: y = 2.5x^2 + 2.5x + 45
:
"use the result to predict the sales during the fourth year."
x = 4
y = 2.5(4^2) + 2.5(4) + 45
y = 2.5(16) + 10 + 45
y = 40 + 10 + 45
y = 95 million in the 4th year
:
:
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