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
To find a quadratic function that fits the given sales data, we need to solve a system of equations using the three data points.
Let's denote the sales (in millions of dollars) as y and the years as x. The three data points can be written as (x, y):
Point 1: (1, 50)
Point 2: (2, 60)
Point 3: (3, 75)
Now, we can substitute these values in the quadratic function equation: y = ax^2 + bx + c.
Substituting the values from Point 1:
50 = a(1^2) + b(1) + c --(1)
Substituting the values from Point 2:
60 = a(2^2) + b(2) + c --(2)
Substituting the values from Point 3:
75 = a(3^2) + b(3) + c --(3)
Now, we have three equations with three unknowns (a, b, and c). We can solve this system of equations using various methods, such as substitution or elimination.
Let's solve it using the substitution method:
From equation (1), we can express c in terms of a and b:
c = 50 - a - b --(4)
Substitute equation (4) into equations (2) and (3):
60 = a(2^2) + b(2) + (50 - a - b)
75 = a(3^2) + b(3) + (50 - a - b)
Simplifying these equations, we get:
60 = 4a + 2b + 50 - a - b --(5)
75 = 9a + 3b + 50 - a - b --(6)
Simplify the above equations further:
10 = 3a + b --(7)
25 = 8a + 2b --(8)
Now, we have a system of two linear equations with two variables. We can solve this system using any method, such as substitution or elimination.
Let's solve it using the elimination method:
Multiply equation (7) by 2 and equation (8) by -1:
20 = 6a + 2b --(9)
-25 = -8a - 2b --(10)
Add equations (9) and (10) to cancel out the b term:
20 - 25 = 6a - 8a
-5 = -2a
Solve for a:
-5 = -2a
a = (-5)/(-2)
a = 2.5
Now, substitute the value of a back into equation (7) to solve for b:
10 = 3a + b
10 = 3(2.5) + b
10 = 7.5 + b
b = 10 - 7.5
b = 2.5
Therefore, we have found the values of a and b: a = 2.5, b = 2.5.
Now, substitute the values of a and b into equation (4) to find c:
c = 50 - a - b
c = 50 - 2.5 - 2.5
c = 45
So, the quadratic function that fits the given data is: y = 2.5x^2 + 2.5x + 45.
To predict the sales during the fourth year (x = 4), substitute x = 4 into the equation:
y = 2.5(4^2) + 2.5(4) + 45
y = 2.5(16) + 10 + 45
y = 40 + 10 + 45
y = 95
Therefore, the predicted sales during the fourth year is 95 million dollars.