The table shows the fuel economy y (mpg) at different speeds x (mph) for a new car prototype.
Speed (mph)
15
25
35
45
55
Fuel Economy (mpg)
21.4
26.6
27.9
29.0
30.5
Use a quadratic function that models the data to estimate the fuel economy for 52 mph.
Your answer:
To find a quadratic function that models the data, we can use the data points given. Let's assume the quadratic function is of the form y = ax^2 + bx + c.
Plugging in the values from the table, we get the following system of equations:
(1) 21.4 = a(15)^2 + b(15) + c
(2) 26.6 = a(25)^2 + b(25) + c
(3) 27.9 = a(35)^2 + b(35) + c
(4) 29.0 = a(45)^2 + b(45) + c
(5) 30.5 = a(55)^2 + b(55) + c
Simplifying the equations, we have the following system:
(1) 225a + 15b + c = 21.4
(2) 625a + 25b + c = 26.6
(3) 1225a + 35b + c = 27.9
(4) 2025a + 45b + c = 29.0
(5) 3025a + 55b + c = 30.5
Using a calculator or solving the system of equations manually, we find:
a ≈ -0.000642857142857
b ≈ 0.1965
c ≈ 18.035
So the quadratic function that models the data is:
y = -0.000642857142857x^2 + 0.1965x + 18.035
To estimate the fuel economy at 52 mph, we can substitute x = 52 into the equation:
y = -0.000642857142857(52)^2 + 0.1965(52) + 18.035
Calculating, we find:
y ≈ 29.725
Therefore, the estimated fuel economy for 52 mph is approximately 29.725 mpg.