dormer window student - mathematics in education and...
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ContentsInitial Problem Statement 2 Narrative 3-9 Appendix 10
Dormer Window ExtensionHow can an engineer determine lengths and angles from the drawings of three-dimensional objects?
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How can an engineer determine lengths and angles from the drawings of three-dimensional objects?
Before an extension or alteration can be made to
a building, an engineer must plan and design the
details. This is done using both two-dimensional
projections and three-dimensional drawings.
Once drawn, the representation allows design
lengths and angles to be determined which can
then be used in the construction.
Dormer Window ExtensionInitial Problem Statement
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Narrative IntroductionThe following drawings show the location and size of a planned dormer window extension to a house. The extension is symmetrical about a vertical centre line on the front view (front elevation). All dimensions are given in mm.
Figure 1
Discussion 1How could you produce such drawings and what might they be used for?
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Activity 1The architect has not yet stated the length x. Use trigonometry to calculate its value in mm to the nearest whole number.
800 800
3100
x
9922
A
A
Figure 2
Discussion 2Is there another method you could use to calculate x?
Activity 2Produce a three-dimensional sketch of what the frame for the extension will look like. Label your sketch with the known lengths.
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Some of the dimensions and angles can be found using more than one method. Identify alternative
calculations and use them to check your results.
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2. ConstructionIn order to construct the extension a hole of the appropriate size and shape needs to be made in the existing roof.
800
800
2500
2561
A
a
b
B
C
Figure 3
The construction will have some engineering tolerances built in so you can use 2561 mm as marked and take A = 32.0°.
Activity 3Sketch the shape of the hole required in the roof. Calculate the dimensions a and b to the nearest mm and the angles B and C to 1 decimal place.
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3. Effects of the extension on the buildingDiscussion 3Adding an extension changes the design of the house. What effects externally and internally might arise as a result of these changes? What might you need to calculate from the drawing to evaluate the magnitude of these effects?
Discussion 4Look at the diagram below. How many exterior plane surfaces are there in the design? How many uniquely shaped exterior plane surfaces are there?
800
800
2500
2561
32°
c
1960.148…
D
E d
Q
R V
W
P Figure 5
Activity 4Sketch the roof section QRVW. Calculate the lengths c and d giving results to the nearest mm value. Calculate the angles D and E, giving results to 1 d.p.
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4. Calculating the surface areaYou have discovered that in order to calculate the total surface area of the extension you need to calculate the area of three unique plane shapes; a roof section, the front and a triangular side. You have also calculated the dimensions of the sides of the plane shapes.
A roof section is shown below. All dimensions are shown in mm and have been rounded to appropriate values for construction allowing for engineering tolerances.
1484
2561
1280.5
1960
2561
1960
1280.5
1484
Figure 7
Figure 8
Discussion 5What would be an appropriate unit for the area?
Activity 5Make an order of magnitude estimate of the area.
Activity 6Establish an approximate upper and lower bound on the area.
Activity 7Calculate the area of the roof section using the above dimensions giving your result in appropriate units, as discussed above.
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Discussion 6What is the effect of rounding the lengths on the calculated area? Recall that the base was previously calculated lengths are 1484.082…, 1280.5 and 1960.148….
Activity 8Calculate the area of the front and side parts as shown below and give the total new external surface area.
800
2500
1600
Figure 9
800
1280.5
1510
Figure 10
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5. Checking your resultsAll of the calculations you have carried out in this problem can be automated using CAD software.
Activity 9Create a CAD model of the building and extension. Use it to check your results.
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Appendixmathematical coverageUse trigonometry and coordinate geometry to solve engineering problems• KnowandbeabletousePythagoras'theorem• Knowthemeaningsofsine,cosineandtangent• Beabletofindsine,cosineandtangentofanyangle• Beabletousesine,cosineandtangenttofindunknownsidesandanglesinright-angledtriangles• Beabletofindsides,anglesandareasinfiguresinvolvingmorethanonetriangle• Solvesimpleequationsinvolvingtrigonometricfunctions• Beabletointerpretdrawingsofthree-dimensionalobjects• Beabletodrawsimplethree-dimensionalobjects• Beabletocalculatelengthsandanglesinthree-dimensionalobjects
Use algebra to solve engineering problems• Be able to use appropriate units• Be able to convert from one set of units to another• Be able to comment on the expected order of magnitude of an answer and to decide if the
answer is reasonable• Be able to establish reasonable upper and lower bounds for the answer to a problem• Work with formulae, including those for the areas and perimeters of plane shapes, and the
surface areas and volumes of regular solids