Aquatic Quadratics Support Sheet
Example Data:
Data Table: Water Level as a
Function of Time
Time (sec)
Water Level (cm)
0
14
30
11.1
60
8.2
90
6.1
120
3.9
150
2.4
180
1.1
210
0.4
240
0
270
0
300
0
Create three equations using three data points from the table (bolded in the example table):
(30, 11.1)
y = ax2 + bx + c
11.1 = a(30)2 + b(30) + c
11.1 = 900a + 30b + c (1)
(120, 3.9)
y = ax2 + bx + c
3.9 = a(120)2 + b(120) + c
3.9 = 14400a + 120b + c (2)
(210, 0.4)
y = ax2 + bx + c
0.4 = a(210)2 + b(210) + c
0.4 = 44100a + 210b + c (3)
Since all three equations contain a single c, multiplying an equation by -1 will allow c to be cancelled out when
two of the equations are added as shown below, forming equation (4).
(2) 14400a + 120b + c = 3.9
(1) 900a + 30b + c = 11.1
14400a + 120b + c = 3.9
-1[900a + 30b + c = 11.1]
14400a + 120b + c = 3.9
-900a - 30b - c = -11.1
13500a + 90b + 0 = -7.2 (4)
Solve (4) for b (in terms of a) and substitute the result for b in (3), then solve for c (in terms of a):
13500a + 90b = -7.2 (4)
90(150a + b) = -7.2
150a + b = -0.08
b = -150a – 0.08
44100a + 210b + c = 0.4 (3)
44100a + 210(-150a – 0.08) + c = 0.4
44100a – 31500a – 16.8+ c = 0.4
12600a – 16.8 + c = 0.4
c = -12600a + 17.2
Insert the solutions for b and c into (1) to find the value of a:
900a + 30b + c = 11.1 (1)
900a + 30(-150a – 0.08) + (-12600a + 17.2) = 11.1
900a – 4500a – 2.4 - 12600a + 17.2 = 11.1
-16200a + 14.8 = 11.1
-16200a = -3.7
a = 0.0002
Insert value of a into the solution for b to find value of b:
b = -150a – 0.08
b = -150(0.0002) – 0.08
b = -0.03 – 0.08
Aquatic Quadratics, Support Sheet, page 1
b = -0.11
© 2013, RAFT
Insert value of a into the solution for c to find value of c:
c = -12600a + 17.2
c = -12600(0.0002) + 17.2
c = -2.52 + 17.2
c = 14.7
Write the mathematical model describing the water level-time relationship by inserting the values for a, b, and c
into y = ax2 + bx + c:
y = ax2 + bx + c
y = 0.0002x2 – 0.11x + 14.7
Insert another x-value from the table into the model and calculate y, then compare to measured y.
x = 180:
y = 0.0002x2 – 0.11x + 14.7
y = 0.0002(180)2 – 0.11(180) + 14.7
y = 0.0002(32400) – 19.8 + 14.7
y = 6.48 – 19.8 + 14.7
y = 1.4, which is close to the measured value of 1.1 cm in the table above.
Aquatic Quadratics, Support Sheet, page 2
© 2013, RAFT