Substitute the values from the table into a quadratic equation in standard form and find the coefficients. Then, substitute 30 into the equation.
y=0.005x^2-1.95x+120, 66mm
Practice makes perfect
We are given a table that shows the water levels at three different times.
Elapsed Time (s)
Water Level (mm)
0
120
20
83
40
50
To find a quadratic function that models the data, we will start by remembering the standard form of a parabola.
y = ax^2 +bx + c
In our case x will represent the time and y will be the water level. To determine the coefficients, we will substitute the values given in the table.
Elapsed Time (s)
Water Level (mm)
y=ax^2 + bx + c
0
120
120 = a(0)^2 + b(0) + c
20
83
83 = a(20)^2 + b(20) + c
40
50
50 = a(40)^2 + b(40) + c
Next, we simplify and rearrange each of the equations above and get the system below.
c=120 & (I) 400a+20b+c=83 & (II) 1600a+40b+c=50 & (III)
From the first equation we already know the value of c, namely, c=120. Let's substitute it into the other two equations and simplify them.
c=120 & (I) 400a+20b=-37 & (II) 1600a+40b=-70 & (III)
Let's multiply Equation (II) by -2 and add the resulting equation to Equation (III).
- 800a - 40b &= 74
^+ 1600a+40b &= - 70
800a &= 4
From the final equation we obtain a=0.005 Let's substitute this value into Equation (II) to find the value of b.
Consequently, the quadratic equation that models the given data is y=0.005x^2-1.95x+120. To estimate the water level at 30 seconds, we substitute x=30 into this equation.
In conclusion, at 30 seconds the water level is 66mm. This is reasonable, since at 20 seconds it was 83mm (greater) and at 40 seconds it was 50mm (fewer).
