Question

an experiment examined the relationship between the number of miles a car traveled, y, per gallon of gasoline and the speed of the car, x, in miles per hour. the table displays the data collected. car mileage experiment table with speed (x) and miles per gallon (y) values: 20, 24.9; 30, 28.3; 35, 29.1; 40, 30.1; 50, 30; 60, 29.1 a quadratic function can be used to model the data in the table. which value best estimates the miles per gallon when the speed is 65 miles per hour? a) 27.9 c) 40.2 b) 31.0 d) 24.9

Explanation:

Step1: Define quadratic model

Let the quadratic function be $y = ax^2 + bx + c$.

Step2: Substitute 3 data points

Use $(20, 24.9)$, $(30, 28.3)$, $(40, 30.1)$:

1. $24.9 = 400a + 20b + c$
2. $28.3 = 900a + 30b + c$
3. $30.1 = 1600a + 40b + c$

Step3: Solve for coefficients

Subtract equation1 from 2: $3.4 = 500a + 10b$ → $0.34 = 50a + b$
Subtract equation2 from 3: $1.8 = 700a + 10b$ → $0.18 = 70a + b$
Subtract new equations: $-0.16 = 20a$ → $a = -0.008$
Substitute $a$: $0.34 = 50(-0.008) + b$ → $b = 0.74$
Substitute $a,b$ into 1: $24.9 = 400(-0.008) + 20(0.74) + c$ → $c = 18.1$
Model: $y = -0.008x^2 + 0.74x + 18.1$

Step4: Calculate at x=65

$y = -0.008(65)^2 + 0.74(65) + 18.1$
$y = -0.008(4225) + 48.1 + 18.1$
$y = -33.8 + 48.1 + 18.1 = 27.4$
This is closest to 27.9.

Answer:

a) 27.9