For depth vs. clock time observations:

Table, based on average of class observations:

clock time position
0 15
4 13
7 11
11 9
14 7
19 5
25 3
31 1

Point graph of depth vs. clock time (note that depth is on vertical axis, clock time on horizontal, as dictated by the language 'depth vs. clock time')::

Your first equation will use t = 5.  If your depth estimate, based on a smooth curve thru the data points (NOT on actual data points), is y = 12 cm then the equation for this first point would be

12 = a * 5^2 + b*5 + c or

If the estimate for the t = 15 depth is 6 cm then we get the equation

Using estimated depth 1 cm for t = 30 sec we get the equation

We now have three simultaneous equations in a, b and c.

We can solve these equations for a, b and c by elimination.

The result is

Plugging these values into the form y = a t^2 + b t + c we get the model

We can graph this equation using Excel, and we can then construct a scatter graph of y vs. t.  If we then right-click on a data point and choose Add Trendline, then choose a polynomial of degree 2 (and also click on Options > Display Equation of Graph), we will obtain a quadratic trendline with the 'best-fit' equation.

The Excel trendline gives us the model:

Excel graph:

If the model is

then at what clock time will the depth be 10 cm?

Plug in 10 for y and you get

We will solve this equation using the Quadratic Formula.

Assignment:  Use the sheets handed out in class, or go to http://vhmthphy.vhcc.edu > Precalculus > Assts and look at Assignments 1 and 2 which give you links to the following:

You can also refer if you need additional explanations to Class Notes #2 under the Lectures link.