MATHEMATICAL MODELING AND PROBLEM SOLVING 5

MathematicalModeling and Problem Solving

Whenusing the polya’s problem solving technique, there are four stepsthat need to be followed in solving a problem. These steps would bediscussed in solving the problem under consideration.

Step1: Understand the Problem

Inthis step, one has to understand what the problem requires by statingthe problem in own words. In the problem under consideration, theamount of paint and the subsequent cost are the unknowns. Theprovided information constitutes the breadth, length and height ofthe room to be painted. Also, information concerning the length andbreadth of the window in the room that should not be painted has beenprovided. Furthermore, the cost and area covered by paint that willbe used for painting the room and the ceiling have also beenprovided. The following is the data

Lengthof the room = 16ft

Breadthof the room = 14ft 3 inch

Heightof the room = 8ft

Lengthof the window = 5 ft

Breadthof the window = 3 ft 4 inch

Areacovered by color used to paint the room = 75 sq. ft per gallon

Costof the color used in painting the room = $33.50 per gallon

Areacovered by paint used to paint the ceiling = 50 sq. ft per gallon

Costof ceiling paint = $28 per gallon

Step2: Devise a Plan

Inthis step, one needs to devise a plan that would be used in gettingthe unknown. In this case, some of the dimensions have been given ininches and they first of all need to be converted to feet so as tomatch with other dimension. Thus, one has to have a plan of how to dothis. In order to convert inches into feet, it is important to havean understanding that 1inch = 0.083333 foot. Also, it is alsoimportant to formulate an equation that would help in getting theunknowns in the problem. The equations that will help in getting theunknowns are as follows

Amountof paint used = total area to be painted divided by area covered bythe paint

Totalcost of paint = cost per gallon multiplied by the number of gallonsof paints used

Step3: Carry Out the Plan

Afterdevising a plan, the third step constitutes carrying out the devisedplan. This will help in establishing the unknowns.

Areaof the bedroom to be painted = 2(H * W) + 2(L * H) – Area of thewindow (This is because it is only the walls of the bedroom that willbe painted and the window will not be painted)

Sincethe width of the bedroom is given in foot and inches, it is importantto convert the inches to foot. Therefore, the width of the bedroomwill be 14 ft + 0.08333*3 = 14.25ft

Widthof the window is also given in foot and inches thus, it is importantto converts the inches to foot. Therefore, the width of the window =3ft + 0.08333 * 4 = 3.33 ft

Areaof the window = 3.33 ft * 5 ft = 16.65 sq. ft

Totalarea of the bedroom to be painted = 2(8 * 14.25) + 2(16 * 8) –16.65 sq. ft

=228 + 256 – 16.65

=467.35 sq. ft

Amountof paint to be used in painting the walls of the bedroom = total areato be painted dividing by area covered by paint for painting thewalls

=467.35 /75

=6.2 gallons

Costof painting the bedroom walls = cost per gallon multiplied by thenumber of gallons

Costof painting bedroom walls = $33.50 * 6.2

=$207.70

Areaof the ceiling = 16 * 14.25 = 228 sq. ft

Amountof paint used to paint the ceiling = 228 /50

=4.56 gallons, which is approximately 5 gallons

Costof painting the ceiling = number of gallons * cost of paint pergallon

=5 * 28

=$ 140

Therefore,the cost of painting the bedroom (ceiling included) will be equal to$207.70 + $140

=$347.70 (this is amount for a single coat)

Hence,in painting the bedroom (ceiling included) with two coats of paint,it will cost 2(347.70)

=$ 695.40

Thetotal area of the bedroom to be painted (ceiling included) will be467.35 + 228, which is 695.35 sq. ft

Giventhat 100 sq. ft can be painted in one hour, then the time taken topaint the bedroom (ceiling included) will be equal to 695.35 sq ftdivided by 100, which will give 6.95 hours. This is approximately 7hours.

Step4: Checking

Thefourth step of the polya’s problem solving techniques entailschecking whether the solution makes sense. In this step, the solutionmust be in a position to make sense and apply logic. In case thesolution does not follow logic, then the solution is not correct. Inthis case, the solution makes sense.