After completing Part $3,$ you should have solved the initial value problem for the beaker of hot water cooling in a room with a controlled fixed temperature. Build a complete model, i$.$e$.$ differential equation, for the rate of change of the temperature of the

water, $T^{\text{'}}(t),$ in terms of $T(t)$ and the function for $T_E(t)$ you determined in #$7.$ Estimating the constant of

proportionality, $k,$ in the differential equation can be challenging, so the $k$ you determined in the

constant room temperature model is at least a good starting point.

Qualitative analysis encompasses techniques for analyzing differential equations and initial$-$value problems

without solving them analytically or numerically.

Create a direction $($or slope$)$ field for the differential equation you have determined, $\frac{dT}{dt}=k(T-T_E(t)),$

where $T_E(t)$ is the function you determined in Part $3.$ Winplot $($you can download the winplot.exe file

from Blackboard in Technology Tools$)$ can draw direction or slope fields for a first order ordinary

differential equation of the form $\frac{dy}{dx}=f(x,y).$ It can also numerically solve initial value problems for

these differential equations.

a$.$ Double click on the winplot icon to open, then perform the sequence of clicks Window $-2-$

dim $-$ Equa $-$ Differential $-$ dy$/$dx to open the differential equation dialog box.

b$.$ Type the formula for $f(x,y)$ in the box. You will enter $k(y-T_E(x)),$ where $T_E(x)$ is the function

you determined in Part $3.$ If you wish, you may use the letter K for the proportionality constant,

then investigate the effect of changing the value of $k$ as explained in $($ g $).$ Make sure that

slopes is checked. You can adjust the lengths and number of rows of slope segments by

changing the numbers in the corresponding boxes $-$ you can also alter the pen widths of these

segments. Click OK to view the slope field.

c$.$ You may adjust the graphing window and add labels and markings, by clicking View $-$ View...

to open the window which allows you change the part of $x-y$ plane you see. I prefer to set

corners. Click that circle, then set left $-10,$ right $100,$ down $-10,$ and up $110.$ Why have

those values been specified? You may also superimpose on the direction$/$slope field the graphs

of one or more equations. $($It is informative to graph solutions of the ordinary differential

equation on top of the slope field, to see how the solution curves follow the field.$)$

d$.$ To solve an initial value problem numerically, first graph the direction$/$slope field as described

Solve your model numerically using the Improved Euler Method and validate your model by comparing

its predictions to the data. If needed, try to improve your estimates of constants in the model.ot running device, the heat was coming on$,$ and the sun was rising. So$,$ they were not surprised that the heat was rising in the room. In Table $1,$ they sample the time, temperature of the water, and temperature of the room or environment. The more complete data table is in CoolltData.xlsx$.$

$\backslash$table$[[$Time $($min$),$Room Temp $(\backslash$deg F$),$Water Temp $(\backslash$deg F$)],[0\mathrm{.}0,,103\mathrm{.}1],[5\mathrm{.}0,58\mathrm{.}3,99\mathrm{.}0],[9\mathrm{.}5,59\mathrm{.}7,96\mathrm{.}0],[15\mathrm{.}0,61\mathrm{.}0,92\mathrm{.}8],[20\mathrm{.}5,62\mathrm{.}1,89\mathrm{.}9],[27\mathrm{.}0,62\mathrm{.}8,86\mathrm{.}9],[35\mathrm{.}0,63\mathrm{.}7,83\mathrm{.}9],[44\mathrm{.}0,64\mathrm{.}2,81\mathrm{.}0],[54\mathrm{.}0,64\mathrm{.}8,78\mathrm{.}4],[58\mathrm{.}5,64\mathrm{.}9,77\mathrm{.}5],[70\mathrm{.}0,65\mathrm{.}5,75\mathrm{.}3],[75\mathrm{.}0,65\mathrm{.}7,74\mathrm{.}6],[80\mathrm{.}0,65\mathrm{.}8,73\mathrm{.}8],[88\mathrm{.}5,66\mathrm{.}0,72\mathrm{.}7],[102\mathrm{.}0,66\mathrm{.}2,71\mathrm{.}4],[106\mathrm{.}5,66\mathrm{.}2,70\mathrm{.}9],[112\mathrm{.}5,66\mathrm{.}4,70\mathrm{.}5],[117\mathrm{.}0,66\mathrm{.}4,70\mathrm{.}2],[126\mathrm{.}0,66\mathrm{.}6,69\mathrm{.}7],[130\mathrm{.}5,66\mathrm{.}7,69\mathrm{.}4],[139\mathrm{.}5,66\mathrm{.}7,69\mathrm{.}0]]$

Table $4.$ Sample of data from the temperature of water $(200mL)$ in a beaker where the environmental temperature is changing.

Model the room temperature of the environment, $T_E(t),$ as a function of time, $t.$ Using Microsoft Excel, with the Solver Add$-$In$,$ may help you determine good parameters for your model.